See Nicolaus Damascenus, quoting from the Lydian historian Xanthus, in C. Müller, Fragmenta historicorum Graecorum, iii.; R. Schubert, Geschichte der Könige von Lydien (1884); M. G. Radet, La Lydie et le monde grec au temps de Mermnades (1892-1893): H. Gelzer, “Das Zeitalter des Gyges” (Rhein. Mus., 1875); H. Winckler, Altorientalische Forschungen, i. (1893); Macan’s edition of Herodotus.

Thucydides vi. 93. 104, vii.; Plutarch, Nicias, 19, 21, 27, 28, Lysander, 16, 17; Diodorus xiii. 7, 8, 28-32; Polyaenus i. 39. 42. See Syracuse (for the siege operations), commentaries on Thucydides and the Greek histories.

See J. L. Heiberg, Peter Andreas Heiberg og Thomasine Gyllembourg (Copenhagen, 1882), and L. Kornelius-Hybel, Nogle Bemaerkninger om P. A. Heiberg og Fru Gyllembourg (Copenhagen, 1883).

See M. Höjer, Öfversigt af Sveriges yttre politik under åren 1676-1680 (Upsala, 1875).

The chief work on the ancient gymnastics is Krause, Gymnastik und Agonistik der Hellenen (1841); of more recent works mention may be made of Jäger, Gymnastik der Hellenen (1881); L. Grasberger, Erziehung und Unterricht im klassischen Altertum (1881); J. P. Mahaffy, Old Greek Education (1883); A. S. Wilkins, National Education in Greece (1873); E. Paz, Histoire de la gymnastique (1886); Wickenhagen, Antike und moderne Gymnastik (1891); Becker-Göll, Charicles ii.; Brugsma, Gymnasiorum apud Graecos descriptio (1855); Petersen, Das Gymnasium der Griechen (1858). See also N. Laisné, Gymnastique pratique (Paris, 1879); Collineau, La Gymnastique (Paris, 1884); L’Hygiène à l’école (Paris, 1889); P. de Coubertin, La Gymnastique utilitaire (Paris, 1905); H. Nissen, Rational Home Gymnastics (Boston, 1903).

Gymnospermae.—Trees or shrubs; leaves vary considerably in size and form. Flowers unisexual, except in a few cases (Gnetales) without a perianth. Monoecious or dioecious. Ovules naked, rarely without carpellary leaves, usually borne on carpophylls, which assume various forms. The single megaspore enclosed in the nucellus is filled with tissue (prothallus) before fertilization, and contains two or more archegonia, consisting usually of a large egg-cell and a small neck, rarely of an egg-cell only and no neck (Gnetum and Welwitschia). Microspore spherical or oval, with or without a bladder-like extension of the exine, containing a prothallus of two or more cells, one of which produces two non-motile or motile male cells. Cotyledons two or several. Secondary xylem and phloem produced by a single cambium, or by successive cambial zones; no true vessels (except in the Gnetales) in the wood, and no companion-cells in the phloem.

There is no doubt that the result of recent research and of work now in progress will be to modify considerably the grouping of the conifers. The family Araucarieae, represented by Araucaria and Agathis, should perhaps be separated as a special class and a rearrangement of other genera more in accord with a natural system of classification will soon be possible; but for the present its twofold subdivision may be retained.

Cycadophyta.—A. Cycadales.—Stems tuberous or columnar, not infrequently branched, rarely epiphytic (Peruvian species of Zamia); fronds pinnate, bi-pinnate in the Australian genus Bowenia. Dioecious; flowers in the form of cones, except the female flowers of Cycas, which consist of a rosette of leaf-like carpels at the apex of the stem. Seeds albuminous, with one integument; the single embryo, usually bearing two partially fused cotyledons, is attached to a long tangled suspensor. Stems and roots increase in diameter by secondary thickening, the secondary wood being produced by one cambium or developed from successive cambium-rings.

The cycads constitute a homogeneous group of a few living members confined to tropical and sub-tropical regions. As a fairly typical and well-known example of the Cycadaceae, a species of the genus Cycas (e.g. C. circinalis, C. revoluta, &c.) is briefly described. The stout columnar stem may reach a height of 20 metres, and a diameter of half a metre; it remains either unbranched or divides near the summit into several short and thick branches, each branch terminating in a crown of long pinnate leaves. The surface of the stem is covered with rhomboidal areas, which represent the persistent bases of foliage- and scale-leaves. In some species of Cycas there is a well-defined alternation of transverse zones on the stem, consisting of larger areas representing foliage-leaf bases, and similar but smaller areas formed by the bases of scale-leaves (F and S, fig. 1). The scale-leaves clothing the terminal bud are linear-lanceolate in form, and of a brown or yellow colour; they are pushed aside as the stem-axis elongates and becomes shrivelled, finally falling off, leaving projecting bases which are eventually cut off at a still lower level. Similarly, the dead fronds fall off, leaving a ragged petiole, which is afterwards separated from the stem by an absciss-layer a short distance above the base. In some species of Cycas the leaf-bases do not persist as a permanent covering to the stem, but the surface 755 is covered with a wrinkled bark, as in Cycas siamensis, which has a stem of unusual form (fig. 2). Small tuberous shoots, comparable on a large scale with the bulbils of Lycopodium Selago, are occasionally produced in the axils of some of the persistent leaf-bases; these are characteristic of sickly plants, and serve as a means of vegetative reproduction. In the genus Cycas the female flower is peculiar among cycads in consisting of a terminal crown of separate leaf-like carpels several inches in length; the apical portion of each carpellary leaf may be broadly triangular in form, and deeply dissected on the margins into narrow woolly appendages like rudimentary pinnae. From the lower part of a carpel are produced several laterally placed ovules, which become bright red or orange on ripening; the bright fleshy seeds, which in some species are as large as a goose’s egg, and the tawny spreading carpels produce a pleasing combination of colour in the midst of the long dark-green fronds, which curve gracefully upwards and outwards from the summit of the columnar stem. In Cycas the stem apex, after producing a cluster of carpellary leaves, continues to elongate and produces more bud-scales, which are afterwards pushed aside as a fresh crown of fronds is developed. The young leaves of Cycas consist of a straight rachis bearing numerous linear pinnae, traversed by a single midrib; the pinnae are circinately coiled like the leaf of a fern (fig. 3). The male flower of Cycas conforms to the type of structure characteristic of the cycads, and consists of a long cone of numerous sporophylls bearing many oval pollen-sacs on their lower faces. The type described serves as a convenient representative of its class. There are eight other living genera, which may be classified as follows:—

Classification.—A. Cycadeae.—Characterized by (a) the alternation of scale- and foliage-leaves (fig. 1) on the branched or unbranched stem; (b) the growth of the main stem through the female flower; (c) the presence of a prominent single vein in the linear pinnae; (d) the structure of the female flower, which is peculiar in not having the form of a cone, but consists of numerous independent carpels, each of which bears two or more lateral ovules. Represented by a single genus, Cycas. (Tropical Asia, Australia, &c.).

B. Zamieae.—The stem does not grow through the female flower; both male and female flowers are in the form of cones. (a) Stangerieae.—Characterized by the fern-like venation of the pinnae, which have a prominent midrib, giving off at a wide angle simple or forked and occasionally anastomosing lateral veins. A single genus, Stangeria, confined to South Africa, (b) Euzamieae.—The pinnae are traversed by several parallel veins. Bowenia, an Australian cycad, is peculiar in having bi-pinnate fronds (fig. 5). The various genera are distinguished from one another by the shape and manner of attachment of the pinnae, the form of the carpellary scales, and to some extent by anatomical characters. Encephalartos (South and Tropical Africa).—Large cones; the carpellary scales terminate in a peltate distal expansion. Macrozamia (Australia).—Similar to Encephalartos except in the presence of a spinous projection from the swollen distal end of the carpels. Zamia (South America, Florida, &c.).—Stem short and often divided into several columnar branches. Each carpel terminates in a peltate head. Ceratozamia (Mexico).—Similar in habit to Macrozamia, but distinguished by the presence of two horn-like spinous processes on the apex of the carpels. Microcycas (Cuba).—Like Zamia, except that the ends of the stamens are flat, while the apices of the carpels are peltate. Dioon (Mexico) (fig. 4).—Characterized by the woolly scale-leaves and carpels; the latter terminate in a thick laminar expansion of triangular form, bearing two placental cushions, on which the ovules are situated. Bowenia (Australia).—Bi-pinnate fronds; stem short and tuberous (fig. 5).

The stems of cycads are often described as unbranched; it is true that in comparison with conifers, in which the numerous branches, springing from the main stem, give a characteristic form to the tree, the tuberous or columnar stem of the Cycadaceae Stem and leaf. constitutes a striking distinguishing feature. Branching, however, occurs not infrequently: in Cycas the tall stem often produces several candelabra-like arms; in Zamia the main axis may break up near the base into several cylindrical branches; in species of Dioon (fig. 4) lateral branches are occasionally produced. The South African Encephalartos frequently produces several branches. Probably the oldest example of this genus in cultivation is in the Botanic Garden of Amsterdam, its age is considered by Professor de Vries to be about two thousand years: although an accurate determination of age is impossible, there is no doubt that many cycads grow very slowly and are remarkable for longevity. The thick armour of petiole-bases enveloping the stem is a characteristic Cycadean feature; in Cycas the alternation of scale-leaves and fronds is more clearly shown than in other cycads; in Encephalartos, Dioon, &c., the persistent scale-leaf bases are almost equal in size to those of the foliage-leaves, and there is no regular alternation of zones such as characterizes some species of Cycas. Another type of stem is illustrated by Stangeria and Zamia, also by a few forms of Cycas (fig. 2), in which the fronds fall off completely, leaving a comparatively smooth stem. The Cyas type of frond, except as regards the presence of a midrib in each pinna, characterizes the cycads generally, except Bowenia and Stangeria. In the monotypic genus Bowenia the large fronds, borne singly on the short and thick stem, are bi-pinnate (fig. 5); the segments, which are broadly ovate or rhomboidal, have several forked spreading veins, and resemble the large pinnules of some species of Adiantum. In Stangeria, also a genus represented by one species (S. paradoxa of South Africa), the long and comparatively broad pinnae, with an entire or irregularly incised margin, are very fern-like, a circumstance which led Kunze to describe the plant in 1835 as a species of the fern Lomaria. In rare cases the pinnae of cycads are lobed or branched: in Dioon spinulosum (Central America) the margin of the segments bears numerous spinous processes; in some species of Encephalartos, e.g. E. horridus, the lamina is deeply lobed; and in a species of the Australian genus Macrozamia, M. heteromera, the narrow pinnae are dichotomously branched almost to the base (fig. 6), and resemble the frond of some species of the fern Schizaea, or the fossil genus Baiera (Ginkgoales). An interesting species of Cycas, C. Micholitzii, has recently been described by Sir William Thiselton-Dyer from Annam, where it was collected by one of Messrs Sanders & Son’s collectors, in which the pinnae instead of being of the usual simple type are 756 dichotomously branched as in Macrozamia heteromera. In Ceratozamia the broad petiole-base is characterized by the presence of two lateral spinous processes, suggesting stipular appendages, comparable, on a reduced scale, with the large stipules of the Marattiaceae among Ferns. The vernation varies in different genera; in Cycas the rachis is straight and the pinnae circinately coiled (fig. 3); in Encephalartos, Dioon, &c., both rachis and segments are straight; in Zamia the rachis is bent or slightly coiled, bearing straight pinnae. The young leaves arise on the stem-apex as conical protuberances with winged borders on which the pinnae appear as rounded humps, usually in basipetal order; the scale-leaves in their young condition resemble fronds, but the lamina remains undeveloped. A feature of interest in connexion with the phylogeny of cycads is the presence of long hairs clothing the scale-leaves, and forming a cap on the summit of the stem-apex or attached to the bases of petioles; on some fossil cycadean plants these outgrowths have the form of scales, and are identical in structure with the ramenta (paleae) of the majority of ferns.

The male flowers of cycads are constructed on a uniform plan, and in all cases consist of an axis bearing crowded, spirally disposed sporophylls. These are often wedge-shaped and angular; in some cases they consist of a short, thick Flower. stalk, terminating in a peltate expansion, or prolonged upwards in the form of a triangular lamina. The sporangia (pollen-sacs), which occur on the under-side of the stamens, are often arranged in more or less definite groups or sori, interspersed with hairs (paraphyses); dehiscence takes place along a line marked out by the occurrence of smaller and thinner-walled cells bounded by larger and thicker-walled elements, which form a fairly prominent cap-like “annulus” near the apex of the sporangium, not unlike the annulus characteristic of the Schizaeaceae among ferns. The sporangial wall, consisting of several layers of cells, encloses a cavity containing numerous oval spores (pollen-grains). In structure a cycadean sporangium recalls those of certain ferns (Marattiaceae, Osmundaceae and Schizaeaceae), but in the development of the spores there are certain peculiarities not met with among the Vascular Cryptogams. With the exception of Cycas, the female flowers are also in the form of cones, bearing numerous carpellary scales. In Cycas revoluta and C. circinalis each leaf-like carpel may produce several laterally attached ovules, but in C. Normanbyana the carpel is shorter and the ovules are reduced to two; this latter type brings us nearer to the carpels of Dioon, in which the flower has the form of a cone, and the distal end of the carpels is longer and more leaf-like than in the other genera of the Zamieae, which are characterized by shorter carpels with thick peltate heads bearing two ovules on the morphologically lower surface. The cones of cycads attain in some cases (e.g. Encephalartos) a considerable size, reaching a length of more than a foot. Cases have been recorded (by Thiselton-Dyer in Encephalartos and by Wieland in Zamia) in which the short carpellary cone-scales exhibit a foliaceous form. It is interesting that no monstrous cycadean cone has been described in which ovuliferous and staminate appendages are borne on the same axis: in the Bennettitales (see Palaeobotany: Mesozoic) flowers were produced bearing on the same axis both androecium and gynoecium.

The pollen-grains when mature consist of three cells, two small and one large cell; the latter grows into the pollen-tube, as in the Coniferales, and from one of the small cells two large ciliated spermatozoids are eventually produced. A Microspores and megaspores. remarkable exception to this rule has recently been recorded by Caldwell, who found that in Microcycas Calocoma the body-cells may be eight or even ten in number and the sperm-cells twice as numerous. One of the most important discoveries made during the latter part of the 19th century was that by Ikeno, a Japanese botanist, who first demonstrated the existence of motile male cells in the genus Cycas. Similar spermatozoids were observed in some species of Zamia by H. J. Webber, and more recent work enables us to assume that all cycads produce ciliated male gametes. Before following the growth of the pollen-grain after pollination, we will briefly describe the structure of a cycadean ovule. An ovule consists of a conical nucellus surrounded by a single integument. At an early stage of development a large cell makes its appearance in the central region of the nucellus; this increases in size and eventually forms three cells; the lowest of these grows vigorously and constitutes the megaspore (embryo-sac), which ultimately absorbs the greater part of the nucellus. The megaspore-nucleus divides repeatedly, and cells are produced from the peripheral region inwards, which eventually fill the spore-cavity with a homogeneous tissue (prothallus); some of the superficial cells at the micropylar end of the megaspore increase in size and divide by a tangential wall into two, an upper cell which gives rise to the short two-celled neck of the archegonium, and a lower cell which develops into a large egg-cell. Each megaspore may contain 2 to 6 archegonia. During the growth of the ovum nourishment is supplied from the contents of the cells immediately surrounding the egg-cell, as in the development of the ovum of Pinus and other conifers. Meanwhile the tissue in the apical region of the nucellus has been undergoing disorganization, which results in the formation of a pollen-chamber (fig. 7, C) immediately above the megaspore. Pollination in cycads has always been described as anemophilous, but according to recent observations by Pearson on South African species it seems probable that, at least in some cases, the pollen is conveyed to the ovules by animal agency. The pollen-grains find their way between the carpophylls, which at the time of pollination are slightly apart owing to the elongation of the internodes of the flower-axis, and pass into the pollen-chamber; the large cell of the pollen-grain grows out into a tube (Pt), which penetrates the nucellar tissue and often branches repeatedly; the pollen-grain itself, with the prothallus-cells, projects freely into the pollen-chamber (fig. 7). The nucleus of the outermost (second) small cell (fig. 7, G) divides, and one of the daughter-nuclei passes out of the cell, and may enter the lowest (first) small cell. The outermost cell, by the division of the remaining nucleus, produces two large spermatozoids (fig. 8, a, a). In Microcycas 16 sperm-cells are produced. In the course of division two bodies appear in the cytoplasm, and behave as centrosomes during the karyokinesis; they gradually become threadlike and coil round each daughter nucleus. This thread gives rise to a spiral ciliated band lying in a depression on the body of each spermatozoid; the large spermatozoids eventually escape from the pollen-tube, and are able to perform ciliary movements in the watery liquid which occurs between the thin papery remnant of nucellar tissue and the archegonial necks. Before fertilization a neck-canal cell is formed by the division of the ovum-nucleus. After the body of a spermatozoid has coalesced with the egg-nucleus the latter divides repeatedly and forms a mass of tissue which grows more vigorously in the lower part of the fertilized ovum, and extends upwards towards the apex of the ovum as a peripheral layer of parenchyma surrounding a central space. By further growth this tissue gives rise to a proembryo, which consists, at the micropylar end, of a sac; the tissue at the chalazal end grows into a long and tangled suspensor, terminating in a mass of cells, which is eventually differentiated into a radicle, plumule and two cotyledons. In the ripe seed the integument assumes the form of a fleshy envelope, succeeded internally by a hard woody shell, internal to which is a thin papery membrane—the apical portion of the nucellus—which is easily dissected out as a conical cap covering the apex of the endosperm. A thorough examination of cycadean seeds has recently been made by Miss Stopes, more particularly with a view to a comparison of their vascular supply with that in Palaeozoic gymnospermous seeds (Flora, 1904). The first leaves borne on the seedling axis are often scale-like, and these are followed by two or more larger laminae, which foreshadow the pinnae of the adult frond.

The anatomical structure of the vegetative organs of recent cycads is of special interest as affording important evidence of relationship with extinct types, and with other groups of recent plants. Brongniart, who Anatomy. was the first to investigate in detail the anatomy of a cycadean stem, recognized an agreement, as regards the secondary wood, with Dicotyledons and Gymnosperms, rather than with Monocotyledons. He drew attention also to certain structural similarities between Cycas and Ginkgo. The main anatomical features of a cycad stem may be summarized as follows: the centre is occupied by a large parenchymatous pith traversed by numerous secretory canals, and in some genera by cauline vascular bundles (e.g. Encephalartos and Macrozamia). In addition to these cauline strands (confined to the stem and not connected with the leaves), collateral bundles are often met with in the pith, which form the vascular supply of terminal flowers borne at intervals on the apex of the stem. These latter bundles may be seen in sections of old stems to pursue a more or less horizontal course, passing outwards through the main woody cylinder. This lateral course is due to the more vigorous growth of the axillary branch formed near the base of each flower, which is a terminal structure, and, except in the female flower of Cycas, puts a limit to the apical growth of the stem. The vigorous lateral branch therefore continues the line of the main axis. The pith is encircled by a cylinder of secondary wood, consisting of single or multiple radial rows of tracheids separated by broad medullary rays composed of large parenchymatous cells; the tracheids bear numerous bordered 757 pits on the radial walls. The large medullary rays give to the wood a characteristic parenchymatous or lax appearance, which is in marked contrast to the more compact wood of a conifer. The protoxylem-elements are situated at the extreme inner edge of the secondary wood, and may occur as small groups of narrow, spirally-pitted elements scattered among the parenchyma which abuts on the main mass of wood. Short and reticulately-pitted tracheal cells, similar to tracheids, often occur in the circummedullary region of cycadean stems. In an old stem of Cycas, Encephalartos or Macrozamia the secondary wood consists of several rather unevenly concentric zones, while in some other genera it forms a continuous mass as in conifers and normal dicotyledons. These concentric rings of secondary xylem and phloem (fig. 9) afford a characteristic cycadean feature. After the cambium has been active for some time producing secondary xylem and phloem, the latter consisting of sieve-tubes, phloem-parenchyma and frequently thick-walled fibres, a second cambium is developed in the pericycle; this produces a second vascular zone, which is in turn followed by a third cambium, and so on, until several hollow cylinders are developed. It has been recently shown that several cambium-zones may remain in a state of activity, so that the formation of a new cambium does not necessarily mark a cessation of growth in the more internal meristematic rings. It occasionally happens that groups of xylem and phloem are developed internally to some of the vascular rings; these are characterized by an inverse orientation of the tissues, the xylem being centrifugal and the phloem centripetal in its development. The broad cortical region, which contains many secretory canals, is traversed by numerous vascular bundles (fig. 9, c) some of which pursue a more or less vertical course, and by frequent anastomoses with one another form a loose reticulum of vascular strands; others are leaf-traces on their way from the stele of the stem to the leaves. Most of these cortical bundles are collateral in structure, but in some the xylem and phloem are concentrically arranged; the secondary origin of these bundles from procambium-strands was described by Mettenius in his classical paper of 1860. During the increase in thickness of a cycadean stem successive layers of cork-tissue are formed by phellogens in the persistent bases of leaves (fig. 9, pd), which increase in size to adapt themselves to the growth of the vascular zones. The leaf-traces of cycads are remarkable both on account of their course and their anatomy. In a transverse section of a stem (fig. 9) one sees some vascular bundles following a horizontal or slightly oblique course in the cortex, stretching for a longer or shorter distance in a direction concentric with the woody cylinder. From each leaf-base two main bundles spread right and left through the cortex of the stem (fig. 9, lt), and as they curve gradually towards the vascular ring they present the appearance of two rather flat ogee curves, usually spoken of as the leaf-trace girdles (fig. 9, lt). The distal ends of these girdles give off several branches, which traverse the petiole and rachis as numerous collateral bundles. The complicated girdle-like course is characteristic of the leaf-traces of most recent cycads, but in some cases, e.g. in Zamia floridana, the traces are described by Wieland in his recent monograph on American fossil cycads (Carnegie Institution Publications, 1906) as possessing a more direct course similar to that in Mesozoic genera. A leaf-trace, as it passes through the cortex, has a collateral structure, the protoxylem being situated at the inner edge of the xylem; when it reaches the leaf-base the position of the spiral tracheids is gradually altered, and the endarch arrangement (protoxylem internal) gives place to a mesarch structure (protoxylem more or less central and not on the edge of the xylem strand). In a bundle examined in the basal portion of a leaf the bulk of the xylem is found to be centrifugal in position, but internally to the protoxylem there is a group of centripetal tracheids; higher up in the petiole the xylem is mainly centripetal, the centrifugal wood being represented by a small arc of tracheids external to the protoxylem and separated from it by a few parenchymatous elements. Finally, in the pinnae of the frond the centrifugal xylem may disappear, the protoxylem being now exarch in position and abutting on the phloem. Similarly in the sporophylls of some cycads the bundles are endarch near the base and mesarch near the distal end of the stamen or carpel. The vascular system of cycadean seedlings presents some features worthy of note; centripetal xylem occurs in the cotyledonary bundles associated with transfusion-tracheids. The bundles from the cotyledons pursue a direct course to the stele of the main axis, and do not assume the girdle-form characteristic of the adult plant. This is of interest from the point of view of the comparison of recent cycads with extinct species (Bennettites), in which the leaf-traces follow a much more direct course than in modern cycads. The mesarch structure of the leaf-bundles is met with in a less pronounced form in the flower peduncles of some cycads. This fact is of importance as showing that the type of vascular structure, which characterized the stems of many Palaeozoic genera, has not entirely disappeared from the stems of modern cycads; but the mesarch bundle is now confined to the leaves and peduncles. The roots of some cycads Roots. resemble the stems in producing several cambium-rings; they possess 2 to 8 protoxylem-groups, and are characterized by a broad pericyclic zone. A common phenomenon in cycads is the production of roots which grow upwards (apogeotropic), and appear as coralline branched structures above the level of the ground; some of the cortical cells of these roots are hypertrophied, and contain numerous filaments of blue-green Algae (Nostocaceae), which live as endoparasites in the cell-cavities.

Ginkgoales.—This class-designation has been recently proposed to give emphasis to the isolated position of the genus Ginkgo (Salisburia) among the Gymnosperms. Ginkgo biloba, the maidenhair tree, has usually been placed by botanists in the Taxeae in the neighbourhood of the yew (Taxus), but the proposal by Eichler in 1852 to institute a special family, the Salisburieae, indicated a recognition of the existence of special characteristics which distinguish the genus from other members of the Coniferae. The discovery by the Japanese botanist Hirase of the development of ciliated spermatozoids in the pollen-tube of Ginkgo, in place of the non-motile male cells of typical conifers, served as a cogent argument in favour of separating the genus from the Coniferales and placing it in a class of its own. In 1712 Kaempfer published a drawing of a Japanese tree, which he described under the name Ginkgo; this term was adopted in 1771 by Linnaeus, who spoke of Kaempfer’s plant as Ginkgo biloba. In 1797 Smith proposed to use the name Salisburia adiantifolia in preference to the “uncouth” genus Ginkgo and “incorrect” specific term biloba. Both names are still in common use. On account of the resemblance of the leaves to those of some species of Adiantum, the appellation maidenhair tree has long been given to Ginkgo biloba. Ginkgo is of special interest on account of its isolated position among existing plants, its restricted geographical distribution, and its great antiquity (see Palaeobotany: Mesozoic). This solitary survivor of an ancient stock is almost extinct, but a few old and presumably wild trees are recorded by travellers in parts of China. Ginkgo is common as a sacred tree in the gardens of temples in the Far East, and often cultivated in North America and Europe. Ginkgo biloba, which may reach a height of over 30 metres, forms a tree of pyramidal shape with a smooth grey bark. The leaves (figs. 10 and 11) have a long, slender petiole terminating in a fan-shaped lamina, which may be entire, divided by a median incision into two wedge-shaped lobes, or subdivided into several narrow segments. The venation is like that of many ferns, e.g. Adiantum; the lowest vein in each half of the lamina follows a course parallel to the edge, and gives off numerous branches, which fork repeatedly as they spread in a palmate manner towards the leaf margin. The foliage-leaves occur either scattered on long shoots of unlimited growth, or at the apex of short shoots (spurs), which may eventually elongate into long shoots.

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The flowers are dioecious. The male flowers (fig. 12), borne in the axil of scale-leaves, consist of a stalked central axis bearing loosely disposed stamens; each stamen consists of a slender filament terminating in a small apical scale, which bears Flowers. usually two, but not infrequently three or four pollen-sacs (fig. 12, C). The axis of the flower is a shoot bearing leaves in the form of stamens. A mature pollen-grain contains a prothallus of 3 to 5 cells (Fig. 13, Pg); the exine extends over two-thirds of the circumference, leaving a thin portion of the wall, which on collapsing produces a longitudinal groove similar to the median depression on the pollen-grain of a cycad. The ordinary type of female flower has the form of a long, naked peduncle bearing a single ovule on either side of the apex (fig. 12), the base of each being enclosed by a small, collar-like rim, the nature of which has been variously interpreted. A young ovule consists of a conical nucellus surrounded by a single integument terminating as a two-lipped micropyle. A large pollen-chamber occupies the apex of the nucellus; immediately below this, two or more archegonia (fig. 13, a) are developed in the upper region of the megaspore, each consisting of a large egg-cell surmounted by two neck-cells and a canal-cell which is cut off shortly before fertilization. After the entrance of the pollen-grain the pollen-chamber becomes roofed over by a blunt protuberance of nucellar tissue. The megaspore (embryo-sac) continues to grow after pollination until the greater part of the nucellus is gradually destroyed; it also gives rise to a vertical outgrowth, which projects from the apex of the megaspore as a short, thick column (fig. 13, e) supporting the remains of the nucellar tissue which forms the roof of the pollen-chamber (fig. 13, c). Surrounding the pitted wall of the ovum there is a definite layer of large cells, no doubt representing a tapetum, which, as in cycads and conifers, plays an important part in nourishing the growing egg-cell. The endosperm detached from a large Ginkgo ovule after fertilization bears a close resemblance to that of a cycad; the apex is occupied by a depression, on the floor of which two small holes mark the position of the archegonia, and the outgrowth from the megaspore apex projects from the centre as a short peg. After pollination the pollen-tube grows into the nucellar tissue, as in cycads, and the pollen-grain itself (fig. 13, Pg) hangs down into the pollen-chamber; two large spirally ciliated spermatozoids are produced, their manner of development agreeing very closely with that of the corresponding cells in Cycas and Zamia. After fertilization the ovum-nucleus divides and cell-formation proceeds rapidly, especially in the lower part of the ovum, in which the cotyledon and axis of the embryo are differentiated; the long, tangled suspensor of the cycadean embryo is not found in Ginkgo. It is often stated that fertilization occurs after the ovules have fallen, but it has been demonstrated by Hirase that this occurs while the ovules are still attached to the tree. The ripe seed, which grows as large as a rather small plum, is enclosed by a thick, fleshy envelope covering a hard woody shell with two or rarely three longitudinal keels. A papery remnant of nucellus lines the inner face of the woody shell, and, as in cycadean seeds, the apical portion is readily separated as a cap covering the summit of the endosperm.

The morphology of the female flowers has been variously interpreted by botanists; the peduncle bearing the ovules has been described as homologous with the petiole of a foliage-leaf and as a shoot-structure, the collar-like envelope at the base of the ovules being referred to as a second integument or arillus, or as the representative of a carpel. The evidence afforded by normal and abnormal flowers appears to be in favour of the following interpretation: The peduncle is a shoot bearing two or more carpels. Each ovule is enclosed at the base by an envelope or collar homologous with the lamina of a leaf; the fleshy and hard coats of the nucellus constitute a single integument. The stalk of an ovule, considerably reduced in normal flowers and much larger in some abnormal flowers, is homologous with a leaf-stalk, with which it agrees in the structure and number of vascular bundles. The facts on which this description is based are derived partly from anatomical evidence, and in part from an account given by a Japanese botanist, Fujii, of several abnormal female flowers; in some cases the collar at the base of an ovule, often described as an arillus, is found to pass gradually into the lamina of a leaf bearing marginal ovules (fig. 14, B). The occurrence of more than two ovules on one peduncle is by no means rare; a particularly striking example is described by Fujii, in which an unusually thick peduncle bearing several stalked ovules terminates in a scaly bud (fig. 14, A, b). The frequent occurrence of more than two pollen-sacs and the equally common occurrence of additional ovules have been regarded by some authors as evidence in favour of the view that ancestral types normally possessed a greater number of these organs than are usually found in the recent species. This Anatomy. view receives support from fossil evidence. Close to the apex of a shoot the vascular bundles of a leaf make their appearance as double strands, and the leaf-traces in the upper part of a shoot have the form of distinct bundles, which in the older part of the shoot form a continuous ring. Each double leaf-trace passes through four internodes before becoming a part of the stele; the double nature of the trace is a characteristic feature. Secretory sacs occur abundantly in the leaf-lamina, where they appear as short lines between the veins; they are abundant also in the cortex and pith of the shoot, in the fleshy integument of the ovule, and elsewhere. The secondary wood of the shoot and root conforms in the main to the coniferous type; in the short shoots the greater breadth of the medullary rays in the more internal part of the xylem recalls the cycadean type. The secondary phloem contains numerous thick-walled fibres, parenchymatous cells, and large sieve-tubes with plates on the radial walls; swollen parenchymatous cells containing crystals are commonly met with in the cortex, pith and medullary-ray tissues. The wood consists of tracheids, with circular bordered pits on their radial walls, and in the late summer wood pits are unusually abundant on the tangential walls. A point of anatomical interest is the occurrence in the vascular bundles of the cotyledons, scale-leaves, and elsewhere of a few centripetally developed tracheids, which give to the xylem-strands a mesarch structure such as characterizes the foliar bundles of cycads. The root is diarch in structure, but additional protoxylem-strands may be present at the base of the main root; the pericycle consists of several layers of cells.

This is not the place to discuss in detail the past history of Ginkgo (see Palaeobotany: Mesozoic). Among Palaeozoic genera there are some which bear a close resemblance to the recent type in the form of the leaves; and petrified Palaeozoic seeds, Geological history. almost identical with those of the maidenhair tree, have been described from French and English localities. During the Triassic and Jurassic periods the genus Baiera—no doubt a representative of the Ginkgoales—was widely spread throughout Europe and in other regions; Ginkgo itself occurs abundantly in Mesozoic and Tertiary rocks, and was a common plant in the Arctic regions as elsewhere during the Jurassic and Lower Cretaceous periods. Some unusually perfect Ginkgo leaves have been found in the Eocene leaf-beds between the lava-flows exposed in the cliffs of Mull (fig. 11). From an evolutionary point of view, it is of interest to note the occurrence of filicinean and cycadean characters in the maidenhair tree. The leaves at once invite a comparison with ferns; the numerous long hairs which form a delicate woolly covering on young leaves recall the hairs of certain ferns, but agree more closely with the long filamentous hairs of recent cycads. The spermatozoids constitute the most striking link with both cycads and ferns. The structure of the seed, the presence of two neck-cells in the archegonia, the late development of the embryo, the partially-fused cotyledons and certain anatomical characters, are features common to Ginkgo and the cycads. The maidenhair tree is one of the most interesting survivals from the past; it represents a type which, in the Palaeozoic era, may have been merged into the extinct class Cordaitales. Through the succeeding ages the Ginkgoales were represented by numerous forms, which gradually became more restricted in their distribution and fewer in number during the Cretaceous and Tertiary periods, terminating at the present day in one solitary survivor.

Coniferales.—Trees and shrubs characterized by a copious branching of the stem and frequently by a regular pyramidal form. Leaves simple, small, linear or short and scale-like, usually persisting for more than one year. Flowers monoecious or dioecious, unisexual, without a perianth, often in the form of cones, but never terminal on the main stem.

The plants usually included in the Coniferae constitute a less homogeneous class than the Cycadaceae. Some authors use the term Coniferae in a restricted sense as including those genera which have the female flowers in the form of cones, External features. the other genera, characterized by flowers of a different type, being placed in the Taxaceae, and often spoken of as Taxads. 759 In order to avoid confusion in the use of the term Coniferae, we may adopt as a class-designation the name Coniferales, including both the Coniferae—using the term in a restricted sense—and the Taxaceae. The most striking characteristic of the majority of the Coniferales is the regular manner of the monopodial branching and the pyramidal shape. Araucaria imbricata, the Monkey-puzzle tree, A. excelsa, the Norfolk Island pine, many pines and firs, cedars and other genera illustrate the pyramidal form. The mammoth redwood tree of California, Sequoia (Wellingtonia) gigantea, which represents the tallest Gymnosperm, is a good example of the regular tapering main stem and narrow pyramidal form. The cypresses afford instances of tall and narrow trees similar in habit to Lombardy poplars. The common cypress (Cupressus sempervirens), as found wild in the mountains of Crete and Cyprus, is characterized by long and spreading branches, which give it a cedar-like habit. A pendulous or weeping habit is assumed by some conifers, e.g. Picea excelsa var. virgata represents a form in which the main branches attain a considerable horizontal extension, and trail themselves like snakes along the ground. Certain species of Pinus, the yews (Taxus) and some other genera grow as bushes, which in place of a main mast-like stem possess several repeatedly-branched leading shoots. The unfavourable conditions in Arctic regions have produced a dwarf form, in which the main shoots grow close to the ground. Artificially induced dwarfed plants of Pinus, Cupressus, Sciadopitys (umbrella pine) and other genera are commonly cultivated by the Japanese. The dying off of older branches and the vigorous growth of shoots nearer the apex of the stem produce a form of tree illustrated by the stone pine of the Mediterranean region (Pinus Pinea), which Turner has rendered familiar in his “Childe Harold’s Pilgrimage” and other pictures of Italian scenery. Conifers are not infrequently seen in which a lateral branch has bent sharply upwards to take the place of the injured main trunk. An upward tendency of all the main lateral branches, known as fastigiation, is common in some species, producing well-marked varieties, e.g. Cephalotaxus pedunculata var. fastigiata; this fastigiate habit may arise as a sport on a tree with spreading branches. Another departure from the normal is that in which the juvenile or seedling form of shoot persists in the adult tree; the numerous coniferous plants known as species of Retinospora are examples of this. The name Retinospora, therefore, does not stand for a true genus, but denotes persistent young forms of Juniperus, Thuja, Cupressus, &c., in which the small scaly leaves of ordinary species are replaced by the slender, needle-like leaves, which stand out more or less at right angles from the branches. The flat branchlets of Cupressus, Thuja (arbor vitae), Thujopsis dolabrata (Japanese arbor vitae) are characteristic of certain types of conifers; in some cases the horizontal extension of the branches induces a dorsiventral structure. A characteristic feature of the genus Agathis (Dammara) the Kauri pine of New Zealand, is the deciduous habit of the branches; these become detached from the main trunk leaving a well-defined absciss-surface, which appears as a depressed circular scar on the stem. A new genus of conifers, Taiwania, has recently been described from the island of Formosa; it is said to agree in habit with the Japanese Cryptomeria, but the cones appear to have a structure which distinguishes them from those of any other genus.

With a few exceptions conifers are evergreen, and retain the leaves for several years (10 years in Araucaria imbricata, 8 to 10 in Picea excelsa, 5 in Taxus baccata; in Pinus the needles usually fall in October of their third year). The larch (Larix) Leaves. sheds its leaves in the autumn, in the Chinese larch (Pseudolarix Kaempferi) the leaves turn a bright yellow colour before falling. In the swamp cypress (Taxodium distichum) the tree assumes a rich brown colour in the autumn, and sheds its leaves together with the branchlets which bear them; deciduous branches occur also in some other species, e.g. Sequoia sempervirens (redwood), Thuja occidentalis, &c. The leaves of conifers are characterized by their small size, e.g. the needle-form represented by Pinus, Cedrus, Larix, &c., the linear flat or angular leaves, appressed to the branches, of Thuja, Cupressus, Libocedrus, &c. The flat and comparatively broad leaves of Araucaria imbricata, A. Bidwillii, and some species of the southern genus Podocarpus are traversed by several parallel veins, as are also the still larger leaves of Agathis, which may reach a length of several inches. In addition to the foliage-leaves several genera also possess scale-leaves of various kinds, represented by bud-scales in Pinus, Picea, &c., which frequently persist for a time at the base of a young shoot which has pushed its way through the yielding cap of protecting scales, while in some conifers the bud-scales adhere together, and after being torn near the base are carried up by the growing axis as a thin brown cap. The cypresses, araucarias and some other genera have no true bud-scales; in some species, e.g. Araucaria Bidwillii, the occurrence of small foliage-leaves, which have functioned as bud-scales, at intervals on the shoots affords a measure of seasonal growth. The occurrence of long and short shoots is a characteristic feature of many conifers. In Pinus the needles occur in pairs, or in clusters of 3 or 5 at the apex of a small and inconspicuous short shoot of limited growth (spur), which is enclosed at its base by a few scale-leaves, and borne on a branch of unlimited growth in the axil of a scale-leaf. In the Californian Pinus monophylla each spur bears usually one needle, but two are not uncommon; it would seem that rudiments of two needles are always produced, but, as a rule, only one develops into a needle. In Sciadopitys similar spurs occur, each bearing a single needle, which in its grooved surface and in the possession of a double vascular bundle bears traces of an origin from two needle-leaves. A peculiarity of these leaves is the inverse orientation of the vascular tissue; each of the two veins has its phloem next the upper and the xylem towards the lower surface of the leaf; this unusual position of the xylem and phloem may be explained by regarding the needle of Sciadopitys as being composed of a pair of leaves borne on a short axillary shoot and fused by their margins (fig. 15, A). Long and short shoots occur also in Cedrus and Larix, but in these genera the spurs are longer and stouter, and are not shed with the leaves; this kind of short shoot, by accelerated apical growth, often passes into the condition of a long shoot on which the leaves are scattered and separated by comparatively long internodes, instead of being crowded into tufts such as are borne on the ends of the spurs. In the genus Phyllocladus (New Zealand, &c.) there are no green foliage-leaves, but in their place flattened branches (phylloclades) borne in the axils of small scale-leaves. The cotyledons are often two in number, but sometimes (e.g. Pinus) as many as fifteen; these leaves are usually succeeded by foliage-leaves in the form of delicate spreading needles, and these primordial leaves are followed, sooner or later, by the adult type of leaf, except in Retinosporas, which retain the juvenile foliage. In addition to the first foliage-leaves and the adult type of leaf, there are often produced leaves which are intermediate both in shape and structure between the seedling and adult foliage. Dimorphism or heterophylly is fairly common. One of the best known examples is the Chinese juniper (Juniperus chinensis), in which branches with spinous leaves, longer and more spreading than the ordinary adult leaf, are often found associated with the normal type of branch. In some cases, e.g. Sequoia sempervirens, the fertile branches bear leaves which are less spreading than those on the vegetative shoots. Certain species of the southern hemisphere genus Dacrydium afford particularly striking instances of heterophylly, e.g. D. Kirkii of New Zealand, in which some branches bear small and appressed leaves, while in others the leaves are much longer and more spreading. A well-known fossil conifer from Triassic strata—Voltzia heterophylla—also illustrates a marked dissimilarity in the leaves of the same shoot. The variation in leaf-form and the tendency of leaves to arrange themselves in various ways on different branches of the same plant are features which it is important to bear in mind in the identification of fossil conifers. In this connexion we may note the striking resemblance between some of the New Zealand Alpine Veronicas, e.g. Veronica Hectori, V. cupressoides, &c. (also Polycladus cupressinus, a Composite), and some of the cypresses and other conifers with small appressed leaves. The long linear leaves of some species of Podocarpus, in which the lamina is traversed by a single vein, recall the pinnae of Cycas; the branches of some Dacrydiums and other forms closely resemble those of lycopods; these superficial resemblances, both between different genera of conifers and between conifers and other plants, coupled with the usual occurrence of fossil coniferous twigs without cones attached to them, render the determination of extinct types a very unsatisfactory and frequently an impossible task.

A typical male flower consists of a central axis bearing numerous spirally-arranged sporophylls (stamens), each of which consists of a slender stalk (filament) terminating distally in a more or less prominent knob or triangular scale, and bearing Flowers. two or more pollen-sacs (microsporangia) on its lower surface. The pollen-grains of some genera (e.g. Pinus) are furnished with bladder-like extensions of the outer wall, which serve as aids to wind-dispersal. The stamens of Araucaria and Agathis are peculiar in bearing several long, and narrow free pollen-sacs; these may be compared with the sporangiophores of the horsetails (Equisetum); in Taxus (yew) the filament is attached to the centre of a large circular distal expansion, which bears several pollen-sacs on its under surface. In the conifers proper the female reproductive organs have the form of cones, which may be styled flowers or inflorescences according to different interpretations of their morphology. In the Taxaceae the flowers have a simpler structure. The female flowers of the Abietineae may be taken as representing a common type. A pine cone reaches maturity in two years; a single year suffices for the full development in Larix and several other genera. The axis of the cone bears numerous spirally disposed flat scales (cone-scales), each of which, if examined in a young cone, is found to be double, and to consist of a lower and an upper portion. The latter is a thin flat scale bearing a median ridge or keel (e.g. Abies), on each side of which is situated an inverted ovule, consisting of a nucellus surrounded by a single integument. As the cone grows in size and becomes woody the lower half of the cone-scale, which we may call the carpellary scale, may remain small, and is so far outgrown by the upper half (seminiferous scale) that it is hardly recognizable in the mature cone. In many species of Abies (e.g. Abies pectinata, &c.) the ripe cone differs from those of Pinus, Picea and Cedrus in the large size of the carpellary scales, which project as conspicuous thin appendages beyond the distal margins of the broader and more woody seminiferous scales; the long carpellary scale is a prominent feature also in the cone of the Douglas pine (Pseudotsuga Douglasii). The female flowers (cones) vary considerably in size; the largest are the more or less spherical cones of Araucaria—a single cone of A. imbricata may produce as many as 300 seeds, one seed to each fertile cone-scale—and the long pendent 760 cones, 1 to 2 ft. in length, of the sugar pine of California (Pinus Lambertiana) and other species. Smaller cones, less than an inch long, occur in the larch, Athrotaxis (Tasmania), Fitzroya (Patagonia and Tasmania), &c. In the Taxodieae and Araucarieae the cones are similar in appearance to those of the Abietineae, but they differ in the fact that the scales appear to be single, even in the young condition; each cone-scale in a genus of the Taxodiinae (Sequoia, &c.) bears several seeds, while in the Araucariinae (Araucaria and Agathis) each scale has one seed. The Cupressineae have cones composed of a few scales arranged in alternate whorls; each scale bears two or more seeds, and shows no external sign of being composed of two distinct portions. In the junipers the scales become fleshy as the seeds ripen, and the individual scales fuse together in the form of a berry. The female flowers of the Taxaceae assume another form; in Microcachrys (Tasmania) the reproductive structures are spirally disposed, and form small globular cones made up of red fleshy scales, to each of which is attached a single ovule enclosed by an integument and partially invested by an arillus; in Dacrydium the carpellary leaves are very similar to the foliage leaves—each bears one ovule with two integuments, the outer of which constitutes an arillus. Finally in the yew, as a type of the family Taxeae, the ovules occur singly at the apex of a lateral branch, enclosed when ripe by a conspicuous red or yellow fleshy arillus, which serves as an attraction to animals, and thus aids in the dispersal of the seeds.

It is important to draw attention to some structural features exhibited by certain cone-scales, in which there is no external sign indicative of the presence of a carpellary and a seminiferous scale. In Araucaria Cookii and some allied species each Morphology of female flower. scale has a small pointed projection from its upper face near the distal end; the scales of Cunninghamia (China) are characterized by a somewhat ragged membranous projection extending across the upper face between the seeds and the distal end of the scale; in the scales of Athrotaxis (Tasmania) a prominent rounded ridge occupies a corresponding position. These projections and ridges may be homologous with the seminiferous scale of the pines, firs, cedars, &c. The simplest interpretation of the cone of the Abietineae is that which regards it as a flower consisting of an axis bearing several open carpels, which in the adult cone may be large and prominent or very small, the scale bearing the ovules being regarded as a placental outgrowth from the flat and open carpel. In Araucaria the cone-scale is regarded as consisting of a flat carpel, of which the placenta has not grown out into the scale-like structure. The seminiferous scale of Pinus, &c., is also spoken of sometimes as a ligular outgrowth from the carpellary leaf. Robert Brown was the first to give a clear description of the morphology of the Abietineous cone in which carpels bear naked ovules; he recognized gymnospermy as an important distinguishing feature in conifers as well as in cycads. Another view is to regard the cone as an inflorescence, each carpellary scale being a bract bearing in its axil a shoot the axis of which has not been developed; the seminiferous scale is believed to represent either a single leaf or a fused pair of leaves belonging to the partially suppressed axillary shoot. In 1869 van Tieghem laid stress on anatomical evidence as a key to the morphology of the cone-scales; he drew attention to the fact that the collateral vascular bundles of the seminiferous scale are inversely orientated as compared with those of the carpellary scale; in the latter the xylem of each bundle is next the upper surface, while in the seminiferous scale the phloem occupies that position. The conclusion drawn from this was that the seminiferous scale (fig. 15, B, Sc) is the first and only leaf of an axillary shoot (b) borne on that side of the shoot, the axis of which is suppressed, opposite the subtending bract (fig. 15, A, B, C, Br). Another view is to apply to the seminiferous scale an explanation similar to that suggested by von Mohl in the case of the double needle of Sciadopitys, and to consider the seed-bearing scale as being made up of a pair of leaves (fig. 15, A, a, a) of an axillary shoot (b) fused into one by their posterior margins (fig. 15, A). The latter view receives support from abnormal cones in which carpellary scales subtend axillary shoots, of which the first two leaves (fig. 15, C, l1, l1) are often harder and browner than the others; forms have been described transitional between axillary shoots, in which the leaves are separate, and others in which two of the leaves are more or less completely fused. In a young cone the seminiferous scale appears as a hump of tissue at the base or in the axil of the carpellary scale, but Celakovský, a strong supporter of the axillary-bud theory, attaches little or no importance to this kind of evidence, regarding the present manner of development as being merely an example of a short cut adopted in the course of evolution, and replacing the original production of a branch in the axil of each carpellary scale. Eichler, one of the chief supporters of the simpler view, does not recognize in the inverse orientation of the vascular bundles an argument in support of the axillary-bud theory, but points out that the seminiferous scale, being an outgrowth from the surface of the carpellary scale, would, like outgrowths from an ordinary leaf, naturally have its bundles inversely orientated. In such cone-scales as show little or no external indication of being double in origin, e.g. Araucaria (fig. 15, D) Sequoia, &c., there are always two sets of bundles; the upper set, having the phloem uppermost, as in the seminiferous scale of Abies or Pinus, are regarded as belonging to the outgrowth from the carpellary scale and specially developed to supply the ovules. Monstrous cones are fairly common; these in some instances lend support to the axillary-bud theory, and it has been said that this theory owes its existence to evidence furnished by abnormal cones. It is difficult to estimate the value of abnormalities as evidence bearing on morphological interpretation; the chief danger lies perhaps in attaching undue weight to them, but there is also a risk of minimizing their importance. Monstrosities at least demonstrate possible lines of development, but when the abnormal forms of growth in various directions are fairly evenly balanced, trustworthy deductions become difficult. The occurrence of buds in the axils of carpellary scales may, however, simply mean that buds, which are usually undeveloped in the axils of sporophylls, occasionally afford evidence of their existence. Some monstrous cones lend no support to the axillary-bud theory. In Larix the axis of the cone often continues its growth; similarly in Cephalotaxus the cones are often proliferous. (In rare cases the proliferated portion produces male flowers in the leaf-axils.) In Larix the carpellary scale may become leafy, and the seminiferous scale may disappear. Androgynous cones may be produced, as in the cone of Pinus rigida (fig. 16), in which the lower part bears stamens and the upper portion carpellary and seminiferous scales. An interesting case has been figured by Masters, in which scales of a cone of Cupressus Lawsoniana bear ovules on the upper surface and stamens on the lower face. One argument that has been adduced in support of the axillary bud theory is derived from the Palaeozoic type Cordaites, in which each ovule occurs on an axis borne in the axil of a bract. The whole question is still unsolved, and perhaps insoluble. It may be that the interpretation of the female cone of the Abietineae as an inflorescence, which finds favour with many botanists, cannot be applied to the cones of Agathis and Araucaria. Without expressing any decided opinion as to the morphology of the double cone-scale of the Abietineae, preference may be felt in favour of regarding the cone-scale of the Araucarieae as a simple carpellary leaf bearing a single ovule. A discussion of this question may be found in a paper on the Araucarieae by Seward and Ford, published in the Transactions of the Royal Society of London (1906). Cordaites is an extinct type which in certain respects resembles Ginkgo, cycads and the Araucarieae, but its agreement with true conifers is probably too remote to justify our attributing much weight to the bearing of the morphology of its female flowers on the interpretation of that of the Coniferae. The greater simplicity of the Eichler theory may prejudice us in its favour; but, on the other hand, the arguments advanced in favour of the axillary-bud theories are perhaps not sufficiently cogent to lead us to accept an explanation based chiefly on the uncertain evidence of monstrosities.

A pollen-grain when first formed from its mother-cell consists of a single cell; in this condition it may be carried to the nucellus of the ovule (e.g. Taxus, Cupressus, &c.), or more usually (Pinus, Larix, &c.) it reaches maturity before the dehiscence Micro-spores and megaspores. of the microsporangium. The nucleus of the microspore divides and gives rise to a small cell within the large cell, a second small cell is then produced; this is the structure of the ripe pollen-grain in some conifers (Taxus, &c.). The large cell grows out as a pollen-tube; the second of the two small cells (body-cell) wanders into the tube, followed by the nucleus of the first small cell (stalk-cell). In Taxus the body-cell eventually divides into two, in which the products of division are of unequal size, the larger constituting the male generative cell, which fuses with the nucleus of the egg-cell. In Juniperus the products of division of the 761 body-cell are equal, and both function as male generative cells. In the Abietineae cell-formation in the pollen-grain is carried farther. Three small cells occur inside the cavity of the microspore; two of them collapse and the third divides into two, forming a stalk-cell and a larger body-cell. The latter ultimately divides in the apex of the pollen-tube into two non-motile generative cells. Evidence has lately been adduced of the existence of numerous nuclei in the pollen-tubes of the Araucarieae, and it seems probable that in this as in several other respects this family is distinguished from other members of the Coniferales. The precise method of fertilization in the Scots Pine was followed by V. H. Blackman, who also succeeded in showing that the nuclei of the sporophyte generation contain twice as many chromosomes as the nuclei of the gametophyte. Other observers have in recent years demonstrated a similar relation in other genera between the number of chromosomes in the nuclei of the two generations. The ovule is usually surrounded by one integument, which projects beyond the tip of the nucellus as a wide-open lobed funnel, which at the time of pollination folds inwards, and so assists in bringing the pollen-grains on to the nucellus. In some conifers (e.g. Taxus, Cephalotaxus, Dacrydium, &c.) the ordinary integument is partially enclosed by an arillus or second integument. It is held by some botanists (Celakovský) that the seminiferous scale of the Abietineae is homologous with the arillus or second integument of the Taxaceae, but this view is too strained to gain general acceptance. In Araucaria and Saxegothaea the nucellus itself projects beyond the open micropyle and receives the pollen-grains direct. During the growth of the cell which forms the megaspore the greater part of the nucellus is absorbed, except the apical portion, which persists as a cone above the megaspore; the partial disorganization of some of the cells in the centre of the nucellar cone forms an irregular cavity, which may be compared with the larger pollen-chamber of Ginkgo and the cycads. In each ovule one megaspore comes to maturity, but, exceptionally, two may be present (e.g. Pinus sylvestris). It has been shown by Lawson that in Sequoia sempervirens (Annals of Botany, 1904) and by other workers in the genera that several megaspores may attain a fairly large size in one prothallus. The megaspore becomes filled with tissue (prothallus), and from some of the superficial cells archegonia are produced, usually three to five in number, but in rare cases ten to twenty or even sixty may be present. In the genus Sequoia there may be as many as sixty archegonia (Arnoldi and Lawson) in one megaspore; these occur either separately or in some parts of the prothallus they may form groups as in the Cupressineae; they are scattered through the prothallus instead of being confined to the apical region as in the majority of conifers. Similarly in the Araucarieae and in Widdringtonia the archegonia are numerous and scattered and often sunk in the prothallus tissue. In Libocedrus decurrens (Cupressineae) Lawson describes the archegonia as varying in number from 6 to 24 (Annals of Botany xxi., 1907). An archegonium consists of a large oval egg-cell surmounted by a short neck composed of one or more tiers of cells, six to eight cells in each tier. Before fertilization the nucleus of the egg-cell divides and cuts off a ventral canal-cell; this cell may represent a second egg-cell. The egg-cells of the archegonia may be in lateral contact (e.g. Cupressineae) or separated from one another by a few cells of the prothallus, each ovum being immediately surrounded by a layer of cells distinguished by their granular contents and large nuclei. During the development of the egg-cell, food material is transferred from these cells through the pitted wall of the ovum. The tissue at the apex of the megaspore grows slightly above the level of the archegonia, so that the latter come to lie in a shallow depression. In the process of fertilization the two male generative nuclei, accompanied by the pollen-tube nucleus and that of the stalk-cell, pass through an open pit at the apex of the pollen-tube into the protoplasm of the ovum. After fertilization the nucleus of the egg divides, the first stages of karyokinesis being apparent even before complete fusion of the male and female nuclei has occurred. The result of this is the production of four nuclei, which eventually take up a position at the bottom of the ovum and become separated from one another by vertical cell-walls; these nuclei divide again, and finally three tiers of cells are produced, four in each tier. In the Abietineae the cells of the middle tier elongate and push the lowest tier deeper into the endosperm; the cells of the bottom tier may remain in lateral contact and produce together one embryo, or they may separate (Pinus, Juniperus, &c.) and form four potential embryos. The ripe albuminous seed contains a single embryo with two or more cotyledons. The seeds of many conifers are provided with large thin wings, consisting in some genera (e.g. Pinus) of the upper cell-layers of the seminiferous scale, which have become detached and, in some cases, adhere loosely to the seed as a thin membrane; the loose attachment may be of use to the seeds when they are blown against the branches of trees, in enabling them to fall away from the wing and drop to the ground. The seeds of some genera depend on animals for dispersal, the carpellary scale (Microcachrys) or the outer integument being brightly coloured and attractive. In some Abietineae (e.g. Pinus and Picea)—in which the cone-scales persist for some time after the seeds are ripe—the cones hang down and so facilitate the fall of the seeds; in Cedrus, Araucaria and Abies the scales become detached and fall with the seeds, leaving the bare vertical axis of the cone on the tree. In all cases, except some species of Araucaria (sect. Colymbea) the germination is epigean. The seedling plants of some Conifers (e.g. Araucaria imbricata) are characterized by a carrot-shaped hypocotyl, which doubtless serves as a food-reservoir.

The roots of many conifers possess a narrow band of primary xylem-tracheids with a group of narrow spiral protoxylem-elements at each end (diarch). A striking feature in the roots of several genera, excluding the Abietineae, is the occurrence Anatomy. of thick and somewhat irregular bands of thickening on the cell-walls of the cortical layer next to the endodermis. These bands, which may serve to strengthen the central cylinder, have been compared with the netting surrounding the delicate wall of an inflated balloon. It is not always easy to distinguish a root from a stem; in some cases (e.g. Sequoia) the primary tetrarch structure is easily identified in the centre of an old root, but in other cases the primary elements are very difficult to recognize. The sudden termination of the secondary tracheids against the pith-cells may afford evidence of root-structure as distinct from stem-structure, in which the radial rows of secondary tracheids pass into the irregularly-arranged primary elements next the pith. The annual rings in a root are often less clearly marked than in the stem, and the xylem-elements are frequently larger and thinner. The primary vascular bundles in a young conifer stem are collateral, and, like those of a Dicotyledon, they are arranged in a circle round a central pith and enclosed by a common endodermis. It is in the nature of the secondary xylem that the Coniferales are most readily distinguished from the Dicotyledons and Cycadaceae; the wood is homogeneous in structure, consisting almost entirely of tracheids with circular or polygonal bordered pits on the radial walls, more particularly in the late summer wood. In many genera xylem-parenchyma is present, but never in great abundance. A few Dicotyledons, e.g. Drimys (Magnoliaceae) closely resemble conifers in the homogeneous character of the wood, but in most cases the presence of large spring vessels, wood-fibres and abundant parenchyma affords an obvious distinguishing feature.

The abundance of petrified coniferous wood in rocks of various ages has led many botanists to investigate the structure of modern genera with a view to determining how far anatomical characters may be used as evidence of generic distinctions. There are a few well-marked types of wood which serve as convenient standards of comparison, but these cannot be used except in a few cases to distinguish individual genera. The genus Pinus serves as an illustration of wood of a distinct type characterized by the absence of xylem-parenchyma, except such as is associated with the numerous resin-canals that occur abundantly in the wood, cortex and medullary rays; the medullary rays are composed of parenchyma and of horizontal tracheids with irregular ingrowths from their walls. In a radial section of a pine stem each ray is seen to consist in the median part of a few rows of parenchymatous cells with large oval simple pits in their walls, accompanied above and below by horizontal tracheids with bordered pits. The pits in the radial walls of the ordinary xylem-tracheids occur in a single row or in a double row, of which the pits are not in contact, and those of the two rows are placed on the same level. The medullary rays usually consist of a single tier of cells, but in the Pinus type of wood broader medullary rays also occur and are traversed by horizontal resin-canals. In the wood of Cypressus, Cedrus, Abies and several other genera, parenchymatous cells occur in association with the xylem-tracheids and take the place of the resin-canals of other types. In the Araucarian type of wood (Araucaria and Agathis) the bordered pits, which occur in two or three rows on the radial walls of the tracheids, are in mutual contact and polygonal in shape, the pits of the different rows are alternate and not on the same level; in this type of wood the annual rings are often much less distinct than in Cupressus, Pinus and other genera. In Taxus, Torreya (California and the Far East) and Cephalotaxus the absence of resin-canals and the presence of spiral thickening-bands on the tracheids constitute well-marked characteristics. An examination of the wood of branches, stems and roots of the same species or individual usually reveals a fairly wide variation in some of the characters, such as the abundance and size of the medullary rays, the size and arrangement of pits, the presence of wood-parenchyma—characters to which undue importance has often been attached in systematic anatomical work. The phloem consists of sieve-tubes, with pitted areas on the lateral as well as on the inclined terminal walls, phloem-parenchyma and, in some genera, fibres. In the Abietineae the phloem consists of parenchyma and sieve-tubes only, but in most other forms tangential rows of fibres occur in regular alternation with the parenchyma and sieve-tubes. The characteristic companion-cells of Angiosperms are represented by phloem-parenchyma cells with albuminous contents; other parenchymatous elements of the bast contain starch or crystals of calcium oxalate. When tracheids occur in the medullary rays of the xylem these are replaced in the phloem-region by irregular parenchymatous cells known as albuminous cells. Resin-canals, which occur abundantly in the xylem, phloem or cortex, are not found in the wood of the yew. Cephalotaxus (Taxeae) is also peculiar in having resin-canals in the pith (cf. Ginkgo). One form of Cephalotaxus is characterized by the presence of short tracheids in the pith, in shape like ordinary parenchyma, but in the possession of bordered pits and lignified walls agreeing with ordinary xylem-tracheids; it is probable that these short tracheids serve as reservoirs for storing rather than for conducting water. The vascular bundle entering the stem from a leaf with a single vein passes by a more or less direct course into the 762 central cylinder of the stem, and does not assume the girdle-like form characteristic of the cycadean leaf-trace. In species of which the leaves have more than one vein (e.g. Araucaria imbricata, &c.) the leaf-trace leaves the stele of the stem as a single bundle which splits up into several strands in its course through the cortex. In the wood of some conifers, e.g. Araucaria, the leaf-traces persist for a considerable time, perhaps indefinitely, and may be seen in tangential sections of the wood of old stems. The leaf-trace in the Coniferales is simple in its course through the stem, differing in this respect from the double leaf-trace of Ginkgo. A detailed account of the anatomical characters of conifers has been published by Professor D. P. Penhallow of Montreal and Dr. Gothan of Berlin which will be found useful for diagnostic purposes. The characters of leaves most useful for diagnostic purposes are the position of the stomata, the presence and arrangement of resin-canals, the structure of the mesophyll and vascular bundles. The presence of hypodermal fibres is another feature worthy of note, but the occurrence of these elements is too closely connected with external conditions to be of much systematic value. A pine needle grown in continuous light differs from one grown under ordinary conditions in the absence of hypodermal fibres, in the absence of the characteristic infoldings of the mesophyll cell-walls, in the smaller size of the resin-canals, &c. The endodermis in Pinus, Picea and many other genera is usually a well-defined layer of cells enclosing the vascular bundles, and separated from them by a tissue consisting in part of ordinary parenchyma and to some extent of isodiametric tracheids; but this tissue, usually spoken of as the pericycle, is in direct continuity with other stem-tissues as well as the pericycle. The occurrence of short tracheids in close proximity to the veins is a characteristic of coniferous leaves; these elements assume two distinct forms—(1) the short isodiametric tracheids (transfusion-tracheids) closely associated with the veins; (2) longer tracheids extending across the mesophyll at right angles to the veins, and no doubt functioning as representatives of lateral veins. It has been suggested that transfusion-tracheids represent, in part at least, the centripetal xylem, which forms a distinctive feature of cycadean leaf-bundles; these short tracheids form conspicuous groups laterally attached to the veins in Cunninghamia, abundantly represented in a similar position in the leaves of Sequoia, and scattered through the so-called pericycle in Pinus, Picea, &c. It is of interest to note the occurrence of precisely similar elements in the mesophyll of Lepidodendron leaves. An anatomical peculiarity in the veins of Pinus and several other genera is the continuity of the medullary rays, which extend as continuous plates from one end of the leaf to the other. The mesophyll of Pinus and Cedrus is characterized by its homogeneous character and by the presence of infoldings of the cell-walls. In many leaves, e.g. Abies, Tsuga, Larix, &c., the mesophyll is heterogeneous, consisting of palisade and spongy parenchyma. In the leaves of Araucaria imbricata, in which palisade-tissue occurs in both the upper and lower part of the mesophyll, the resin-canals are placed between the veins; in some species of Podocarpus (sect. Nageia) a canal occurs below each vein; in Tsuga, Torreya, Cephalotaxus, Sequoia, &c., a single canal occurs below the midrib; in Larix, Abies, &c., two canals run through the leaf parallel to the margins. The stomata are frequently arranged in rows, their position being marked by two white bands of wax on the leaf-surface.

The chief home of the Coniferales is in the northern hemisphere, where certain species occasionally extend into the Arctic circle and penetrate beyond the northern limit of dicotyledonous trees. Wide areas are often exclusively occupied by Distribution. conifers, which give the landscape a sombre aspect, suggesting a comparison with the forest vegetation of the Coal period. South of the tree-limit a belt of conifers stretches across north Europe, Siberia and Canada. In northern Europe this belt is characterized by such species as Picea excelsa (spruce), which extends south to the mountains of the Mediterranean region; Pinus sylvestris (Scottish fir), reaching from the far north to western Spain, Persia and Asia Minor; Juniperus communis, &c. In north Siberia Pinus Cembra (Cembra or Arolla Pine) has a wide range; also Abies sibirica (Siberian silver fir), Larix sibirica and Juniperus Sabina (savin). In the North American area Picea alba, P. nigra, Larix americana, Abies balsamea (balsam fir), Tsuga canadensis (hemlock spruce), Pinus Strobus (Weymouth pine), Thuja occidentalis (white cedar), Taxus canadensis are characteristic species. In the Mediterranean region occur Cupressus sempervirens, Pinus Pinea (stone pine), species of juniper, Cedrus atlantica, C. Libani, Callitris quadrivalvis, Pinus montana, &c. Several conifers of economic importance are abundant on the Atlantic side of North America—Juniperus virginiana (red cedar, used in the manufacture of lead pencils, and extending as far south as Florida), Taxodium distichum (swamp cypress), Pinus rigida (pitch pine), P. mitis (yellow pine), P. taeda, P. palustris, &c. On the west side of the American continent conifers play a still more striking rôle; among them are Chamaecyparis nutkaensis, Picea sitchensis, Libocedrus decurrens, Pseudotsuga Douglasii (Douglas fir), Sequoia sempervirens, S. gigantea (the only two surviving species of this generic type are now confined to a few localities in California, but were formerly widely spread in Europe and elsewhere), Pinus Coulteri, P. Lambertiana, &c. Farther south, a few representatives of such genera as Abies, Cupressus, Pinus and juniper are found in the Mexican Highlands, tropical America and the West Indies. In the far East conifers are richly represented; among them occur Pinus densiflora, Cryptomeria japonica, Cephalotaxus, species of Abies, Larix, Thujopsis, Sciadopitys verticillata, Pseudolarix Kaempferi, &c. In the Himalaya occur Cedrus deodara, Taxus, species of Cupressus, Pinus excelsa, Abies Webbiana, &c. The continent of Africa is singularly poor in conifers. Cedrus atlantica, a variety of Abies Pinsapo, Juniperus thurifera, Callitris quadrivalvis, occur in the north-west region, which may be regarded as the southern limit of the Mediterranean region. The greater part of Africa north of the equator is without any representatives of the conifers; Juniperus procera flourishes in Somaliland and on the mountains of Abyssinia; a species of Podocarpus occurs on the Cameroon mountains, and P. milanjiana is widely distributed in east tropical Africa. Widdringtonia Whytei, a species closely allied to W. juniperoides of the Cedarberg mountains of Cape Colony, is recorded from Nyassaland and from N.E. Rhodesia; while a third species, W. cupressoides, occurs in Cape Colony. Podocarpus elongata and P. Thunbergii (yellow wood) form the principal timber trees in the belt of forest which stretches from the coast mountains of Cape Colony to the north-east of the Transvaal. Libocedrus tetragona, Fitzroya patagonica, Araucaria brasiliensis, A. imbricata, Saxegothaea and others are met with in the Andes and other regions in South America. Athrotaxis and Microcachrys are characteristic Australian types. Phyllocladus occurs also in New Zealand, and species of Dacrydium, Araucaria, Agathis and Podocarpus are represented in Australia, New Zealand and the Malay regions.

Gnetales.—These are trees or shrubs with simple leaves. The flowers are dioecious, rarely monoecious, provided with one or two perianths. The wood is characterized by the presence of vessels in addition to tracheids. There are no resin-canals. The three existing genera, usually spoken of as members of the Gnetales, differ from one another more than is consistent with their inclusion in a single family; we may therefore better express their diverse characters by regarding them as types of three separate families—(1) Ephedroideae, genus Ephedra; (2) Welwitschioideae, genus Welwitschia; (3) Gnetoideae, genus Gnetum. Our knowledge of the Gnetales leaves much to be desired, but such facts as we possess would seem to indicate that this group is of special importance as foreshadowing, more than any other Gymnosperms, the Angiospermous type. In the more heterogeneous structure of the wood and in the possession of true vessels the Gnetales agree closely with the higher flowering plants. It is of interest to note that the leaves of Gnetum, while typically Dicotyledonous in appearance, possess a Gymnospermous character in the continuous and plate-like medullary rays of their vascular bundles. The presence of a perianth is a feature suggestive of an approach to the floral structure of Angiosperms; the prolongation of the integument furnishes the flowers with a substitute for a stigma and style. The genus Ephedra, with its prothallus and archegonia, which are similar to those of other Gymnosperms, may be safely regarded as the most primitive of the Gnetales. In Welwitschia also the megaspore is filled with prothallus-tissue, but single egg-cells take the place of archegonia. In certain species of Gnetum described by Karsten the megaspore contains a peripheral layer of protoplasm, in which scattered nuclei represent the female reproductive cells; in Gnetum Gnemon a similar state of things exists in the upper half of the megaspore, while the lower half agrees with the megaspore of Welwitschia in being full of prothallus-tissue, which serves merely as a reservoir of food. Lotsy has described the occurrence of special cells at the apex of the prothallus of Gnetum Gnemon, which he regards as imperfect archegonia (fig. 17, C, a); he suggests they may represent vestigial structures pointing back to some ancestral form beyond the limits of the present group. The Gnetales probably had a separate origin from the other Gymnosperms; they carry us nearer to the Angiosperms, but we have as yet no satisfactory evidence that they represent a stage in the direct line of Angiospermic evolution. It is not improbable that the three genera of this ancient phylum survive as types of a blindly-ending branch of the Gymnosperms; but be that as it may, it is in the Gnetales more than in any other Gymnosperms that we find features which help us to obtain a dim prospect of the lines along which the Angiosperms may have been evolved.

Ephedra.—This genus is the only member of the Gnetales represented in Europe. Its species, which are characteristic of warm temperate latitudes, are usually much-branched shrubs. The finer branches are green, and bear a close resemblance to the stems of Equisetum and to the slender twigs of Casuarina; the surface of the long internodes is marked by fine longitudinal ribs, and at the nodes are borne pairs of inconspicuous scale-leaves. The flowers are small, and borne on axillary shoots. A single male flower consists of an axis enclosed at the base by an inconspicuous perianth formed of two concrescent leaves and terminating in two, or as many as eight, shortly stalked or sessile anthers. The female flower is enveloped in a closely fitting sac-like investment, which must be regarded as a perianth; within this is an orthotropous ovule surrounded by a single integument prolonged upwards as a beak-like micropyle. The flower may be described as a bud bearing a pair of leaves which become fused and constitute a perianth, the apex of the shoot forming an ovule. In function the perianth may be compared with a unilocular ovary containing a single ovule; the projecting integument, which at the time of pollination secretes a drop of liquid, serves the same purpose as the style and stigma of an angiosperm. The megaspore 763 is filled with tissue as in typical Gymnosperms, and from some of the superficial cells 3 to 5 archegonia are developed, characterized by long multicellular necks. The archegonia are separated from one another, as in Pinus, by some of the prothallus-tissue, and the cells next the egg-cells (tapetal layer) contribute food-material to their development. After fertilization, some of the uppermost bracts below each flower become red and fleshy; the perianth develops into a woody shell, while the integument remains membranous. In some species of Ephedra, e.g. E. altissima, the fertilized eggs grow into tubular proembryos, from the tip of each of which embryos begin to be developed, but one only comes to maturity. In Ephedra helvetica, as described by Jaccard, no proembryo or suspensor is formed; but the most vigorous fertilized egg, after undergoing several divisions, becomes attached to a tissue, termed the columella, which serves the purpose of a primary suspensor; the columella appears to be formed by the lignification of certain cells in the central region of the embryo-sac. At a later stage some of the cells in the upper (micropylar) end of the embryo divide and undergo considerable elongation, serving the purpose of a secondary suspensor. The secondary wood of Ephedra consists of tracheids, vessels and parenchyma; the vessels are characterized by their wide lumen and by the large simple or slightly-bordered pits on their oblique end-walls.

Gnetum.—This genus is represented by several species, most of which are climbing plants, both in tropical America and in warm regions of the Old World. The leaves, which are borne in pairs at the tumid nodes, are oval in form and have a Dicotyledonous type of venation. The male and female inflorescences have the form of simple or paniculate spikes. The spike of an inflorescence bears whorls of flowers at each node in the axils of concrescent bracts accompanied by numerous sterile hairs (paraphyses); in a male inflorescence numerous flowers occur at each node, while in a female inflorescence the number of flowers at each node is much smaller. A male flower consists of a single angular perianth, through the open apex of which the flower-axis projects as a slender column terminating in two anthers. The female flowers, which are more complex in structure, are of two types, complete and incomplete; the latter occur in association with male flowers in a male inflorescence. A complete female flower consists of a nucellus (fig. 17, A, n), surrounded by a single integument (fig. 17, A, i), prolonged upwards as a narrow tube and succeeded by an inner and an outer perianth (fig. 17, A, p′ and p″). The whole flower may be looked upon as an adventitious bud bearing two pairs of leaves; each pair becomes concrescent and forms a perianth, the apex of the shoot being converted into an orthotropous ovule. The incomplete female flowers are characterized by the almost complete suppression of the inner perianth. Several embryo-sacs (megaspores) are present in the nucellus of a young ovule, but one only attains full size, the smaller and partially developed megaspores (fig. 17, B and C, e) being usually found in close association with the surviving and fully-grown megaspore. In Gnetum Gnemon, as described by Lotsy, a mature embryo-sac contains in the upper part a large central vacuole and a peripheral layer of protoplasm, including several nuclei, which take the place of the archegonia of Ephedra; the lower part of the embryo-sac, separated from the upper by a constriction, is full of parenchyma. The upper part of the megaspore may be spoken of as the fertile half (fig. 17, B and C, F) and the lower part, which serves only as food-reservoir for the growing embryo, may be termed the sterile half (fig. 17, B and C, S). (Coulter, Bot. Gazette, xlvi., 1908, regards this tissue as belonging to the nucellus.) At the time of pollination the long tubular integument secretes a drop of fluid at its apex, which holds the pollen-grains, brought by the wind, or possibly to some extent by insect agency, and by evaporation these are drawn on to the top of the nucellus, where partial disorganization of the cells has given rise to an irregular pollen-chamber (fig. 17, A, pc). The pollen-tube, containing two generative and one vegetative nucleus, pierces the wall of the megaspore and then becomes swollen (fig. 17, B and C, pt); finally the two generative nuclei pass out of the tube and fuse with two of the nuclei in the fertile half of the megaspore. As the result of fertilization, the fertilized nuclei of the megaspore become surrounded by a cell-wall, and constitute zygotes, which may attach themselves either to the wall of the megaspore or to the end of a pollen-tube (fig. 17, C, z and z′); they then grow into long tubes or proembryos, which make their way towards the prothallus (C, z′), and eventually embryos are formed from the ends of the proembryo tubes. One embryo only comes to maturity. The embryo of Gnetum forms an out-growth from the hypocotyl, which serves as a feeder and draws nourishment from the prothallus. The fleshy outer portion of the seed is formed from the outer perianth, the woody shell being derived from the inner perianth. The climbing species of Gnetum are characterized by the production of several concentric cylinders of secondary wood and bast, the additional cambium-rings being products of the pericycle, as in Cycas and Macrozamia. The structure of the wood agrees in the main with that of Ephedra.

Welwitschia (Tumboa).—This is by far the most remarkable member of the Gnetales, both as regards habit and the form of its flowers. In a supplement to the systematic work of Engler and Prantl the well-known name Welwitschia, instituted by Hooker in 1864 in honour of Welwitsch, the discoverer of the plant, is superseded by that of Tumboa, originally suggested by Welwitsch. The genus is confined to certain localities in Damaraland and adjoining territory on the west coast of tropical South Africa. A well-grown plant projects less than a foot above the surface of the ground; the stem, which may have a circumference of more than 12 ft., terminates in a depressed crown resembling a circular table with a median groove across the centre and prominent broad ridges concentric with the margin. The thick tuberous stem becomes rapidly narrower, and passes gradually downwards into a tap-root. A pair of small strap-shaped leaves succeed the two cotyledons of the seedling, and persist as the only leaves during the life of the plant; they retain the power of growth in their basal portion, which is sunk in a narrow groove near the edge of the crown, and the tough lamina, 6 ft. in length, becomes split into narrow strap-shaped or thong-like strips which trail on the ground. Numerous circular pits occur on the concentric ridges of the depressed and wrinkled crown, marking the position of former inflorescences borne in the leaf-axil at different stages in the growth of the plant. An inflorescence has the form of a dichotomously-branched cyme bearing small erect cones; those containing the female flowers attain the size of a fir-cone, and are scarlet in colour. Each cone consists of an axis, on which numerous broad and thin bracts are arranged in regular rows; in the axil of each bract occurs a single flower; a male flower is enclosed by two opposite pairs of leaves, forming a perianth surrounding a central sterile ovule encircled by a ring of stamens united below, but free distally as short filaments, each of which terminates in a trilocular anther. The integument of the sterile ovule is prolonged above the nucellus as a spirally-twisted tube expanded at its apex into a flat stigma-like organ. A complete and functional female flower consists of a single ovule with two integuments, the inner of which is prolonged into a narrow tubular micropyle, like that in the flower of Gnetum. The megaspore of Welwitschia is filled with a prothallus-tissue before fertilization, and some of the prothallus-cells function as egg-cells; these grow upwards as long tubes into the apical region of the nucellus, where they come into contact with the pollen-tubes. After the egg-cells have been fertilized by the non-motile male cells they grow into tubular proembryos, producing terminal embryos. The stem is traversed by numerous collateral bundles, which have a limited growth, and are constantly replaced by new bundles developed from strands of secondary meristem. One of the best-known anatomical characteristics of the genus is the occurrence of numerous spindle-shaped or branched fibres with enormously-thickened walls studded with crystals of calcium oxalate. Additional information has been published by Professor Pearson of Cape Town based on material collected in Damaraland in 1904 and 1906-1907. In 1906 he gave an account of the early stages of development of the male and female organs and, among other interesting statements in regard to the general biology of Welwitschia, he expressed the opinion that, as Hooker suspected, the ovules are pollinated by insect-agency. In a later paper Pearson considerably extended our knowledge of the reproduction and gametophyte of this genus.

Authorities.—General: Bentham and Hooker, Genera Plantarum (London, 1862-1883); Engler and Prantl, Die natürlichen Pflanzenfamilien (Leipzig, 1889 and 1897); Strasburger, Die Coniferen und Gnetaceen (Jena, 1872); Die Angiospermen und die Gymnospermen (Jena, 1879); Histologische Beiträge, iv. (Jena, 1892); Coulter and Chamberlain, Morphology of Spermatophytes (New York, 1901); Rendle, The Classification of Flowering Plants, vol. i. (Cambridge, 1904); “The Origin of Gymnosperms” (A discussion at the Linnean Society; New Phytologist, vol. v., 1906). Cycadales: Mettenius, “Beiträge zur Anatomie der Cycadeen,” Abh. k. sächs. 764 Ges. Wiss. (1860); Treub, “Recherches sur les Cycadées,” Ann. Bot. Jard. Buitenzorg, ii. (1884); Solms-Laubach, “Die Sprossfolge der Stangeria, &c.,” Bot. Zeit. xlviii. (1896); Worsdell, “Anatomy of Macrozamia,” Ann. Bot. x. (1896) (also papers by the same author, Ann. Bot., 1898, Trans. Linn. Soc. v., 1900); Scott, “The Anatomical Characters presented by the Peduncle of Cycadaceae,” Ann. Bot. xi. (1897); Lang, “Studies in the Development and Morphology of Cycadean Sporangia, No. I.,” Ann. Bot. xi. (1897); No. II., Ann. Bot. xiv. (1900); Webber, “Development of the Antherozoids of Zamia,” Bot. Gaz. (1897); Ikeno, “Untersuchungen über die Entwickelung, &c., bei Cycas revoluta,” Journ. Coll. Sci. Japan, xii. (1898); Wieland, “American Fossil Cycads,” Carnegie Institution Publication (1906); Stopes, “Beiträge zur Kenntnis der Fortpflanzungsorgane der Cycadeen,” Flora (1904); Caldwell, “Microcycas Calocoma,” Bot. Gaz. xliv., 1907 (also papers on this and other Cycads in the Bot. Gaz., 1907-1909); Matte, Recherches sur l’appareil libéro-ligneux des Cycadacées (Caen, 1904). Ginkgoales: Hirase, “Études sur la fécondation, &c., de Ginkgo biloba,” Journ. Coll. Sci. Japan, xii. (1898); Seward and Gowan, “Ginkgo biloba,” Ann. Bot. xiv. (1900) (with bibliography); Ikeno, “Contribution à l’étude de la fécondation chez le Ginkgo biloba,” Ann. Sci. Nat. xiii. (1901); Sprecher, Le Ginkgo biloba (Geneva, 1907). Coniferales: “Report of the Conifer Conference” (1891) Journ. R. Hort. Soc. xiv. (1892); Beissner, Handbuch der Nadelholzkunde (Berlin, 1891); Masters, “Comparative Morphology of the Coniferae,” Journ. Linn. Soc. xxvii. (1891); ibid. (1896), &c.; Penhallow, “The Generic Characters of the North American Taxaceae and Coniferae,” Proc. and Trans. R. Soc. Canada, ii. (1896); Blackman, “Fertilization in Pinus sylvestris,” Phil. Trans. (1898) (with bibliography); Worsdell, “Structure of the Female Flowers in Conifers,” Ann. Bot. xiv. (1900) (with bibliography); ibid. (1899); Veitch, Manual of the Coniferae (London, 1900); Penhallow, “Anatomy of North American Coniferales,” American Naturalist (1904); Engler and Pilger, Das Pflanzenreich, Taxaceae (1903); Seward and Ford, “The Araucarieae, recent and extinct,” Phil. Trans. R. Soc. (1906) (with bibliography); Lawson, “Sequoia sempervirens,” Annals of Botany (1904); Robertson, “Torreya Californica,” New Phytologist (1904); Coker, “Gametophyte and Embryo of Taxodium,” Bot. Gazette (1903); E. C. Jeffrey, “The Comparative Anatomy and Phylogeny of the Coniferales, part i. The Genus Sequoia,” Mem. Boston Nat. Hist. Soc. v. No. 10 (1903); Gothan, “Zur Anatomie lebender und fossiler Gymnospermen-Hölzer,” K. Preuss. Geol. Landes. (Berlin, 1905) (for more recent papers, see Ann. Bot., New Phytologist, and Bot. Gazette, 1906-1909). Gnetales: Hooker, “On Welwitschia mirabilis.” Trans. Linn. Soc. xxiv. (1864); Bower, “Germination, &., in Gnetum,” Journ. Mic. Sci. xxii. (1882); ibid. (1881); Jaccard, “Recherches embryologiques sur l’Ephedra helvetica,” Diss. Inaug. Lausanne (1894); Karsten, “Zur Entwickelungsgeschichte der Gattung Gnetum,” Cohn’s Beiträge, vi. (1893); Lotsy, “Contributions to the Life-History of the genus Gnetum,” Ann. Bot. Jard. Buitenzorg, xvi. (1899); Land, “Ephedra trifurca,” Bot. Gazette (1904); Pearson, “Some observations on Welwitschia mirabilis,” Phil. Trans. R. Soc. (1906); Pearson, “Further Observations on Welwitschia,” Phil. Trans. R. Soc. vol. 200 (1909).

Menstruation.—Normal menstruation comprises the escape of from 4 to 6 oz. of blood together with mucus from the uterus at intervals of twenty-eight days (more or less). The flow begins at the age of puberty, the average age of which in England is between fourteen and sixteen years. It ceases between forty-five and fifty years of age, and this is called the menopause or climacteric period, commonly spoken of as “the change of life.” Both the age of puberty and that of the menopause may supervene earlier or later according to local conditions. At both times the menstrual flow may be replaced by haemorrhage from distant organs (epistaxis, haematemesis, haemoptysis); this is called vicarious menstruation. Menstruation is usually but not necessarily coincident with ovulation. The usual 765 disorders of menstruation are: (1) amenorrhoea (absence of flow), (2) dysmenorrhoea (painful flow), (3) menorrhagia (excessive flow), (4) metrorrhagia (excessive and irregular flow). Amenorrhoea may arise from physiological causes, such as pregnancy, lactation, the menopause; constitutional causes, such as phthisis, anaemia and chlorosis, febrile disorders, some chronic intoxications, such as morphinomania, and some forms of cerebral disease; local causes, which include malformations or absence of one or more of the genital parts, such as absence of ovaries, uterus or vagina, atresia of vagina, imperforate cervix, disease of the ovaries, or sometimes imperforate hymen. The treatment of amenorrhoea must be directed towards the cause. In anaemia and phthisis menstruation often returns after improvement in the general condition, with good food and good sanitary conditions, an outdoor life and the administration of iron or other tonics. In local conditions of imperforate hymen, imperforate cervix or ovarian disease, surgical interference is necessary. Amenorrhoea is permanent when due to absence of the genital parts. The causes of dysmenorrhoea are classified as follows: (1) ovarian, due to disease of the ovaries or Fallopian tubes; (2) obstructive, due to some obstacle to the flow, as stenosis, flexions and malpositions of the uterus, or malformations; (3) congestive, due to subinvolution, chronic inflammation of the uterus or its lining membrane, fibroid growths and polypi of the uterus, cardiac or hepatic disease; (4) neuralgic; (5) membranous. The foremost place in the treatment of dysmenorrhoea must be given to aperients and purgatives administered a day or two before the period is expected. By this means congestion is reduced. Hot baths are useful, and various drugs such as hyoscyanus, cannabis indica, phenalgin, ammonol or phenacetin have been prescribed. Medicinal treatment is, however, only palliative, and flexions and malpositions of the uterus must be corrected, stenosis treated by dilatation, fibroid growths if present removed, and endometritis when present treated by local applications or curetting according to its severity. Menorrhagia signifies excessive bleeding at the menstrual periods. Constitutional causes are purpura, haemophilia, excessive food and alcoholic drinks and warm climates; while local causes are congestion and displacements of the uterus, endometritis, subinvolution, retention of the products of conception, new growths in the uterus such as mucous and fibroid polypi, malignant growths, tubo-ovarian inflammation and some ovarian tumours. Metrorrhagia is a discharge of blood from the uterus, independent of menstruation. It always arises from disease of the uterus or its appendages. Local causes are polypi, retention of the products of conception, extra uterine gestation, haemorrhages in connexion with pregnancy, and new growths in the uterus. In the treatment of both menorrhagia and metrorrhagia the local condition must be carefully ascertained. When pregnancy has been excluded, and constitutional causes treated, efforts should be made to relieve congestion. Uterine haemostatics, as ergot, ergotin, tincture of hydrastis or hamamelis, are of use, together with rest in bed. Fibroid polypi and other new growths must be removed. Irregular bleeding in women over forty years of age is frequently a sign of early malignant disease, and should on no account be neglected.

Diseases of the External Genital Organs.—The vulva comprises several organs and structures grouped together for convenience of description (see Reproductive System). The affections to which these structures are liable may be classified as follows: (1) Injuries to the vulva, either accidental or occurring during parturition; these are generally rupture of the perinaeum. (2) Vulvitis. Simple Vulvitis is due to want of cleanliness, or irritating discharges, and in children may result from threadworms. The symptoms are heat, itching and throbbing, and the parts are red and swollen. The treatment consists of rest, thorough cleanliness and fomentations. Infective vulvitis is nearly always due to gonorrhoea. The symptoms are the same as in simple vulvitis, with the addition of mucopurulent yellow discharge and scalding pain on micturition; if neglected, extension of the disease may result. The treatment consists of rest in bed, warm medicated baths several times a day or fomentations of boracic acid. The parts must be kept thoroughly clean and discharges swabbed away. Diphtheritic vulvitis occasionally occurs, and erysipelas of the vulva may follow wounds, but since the use of antiseptics is rarely seen. (3) Vascular disturbances may occur in the vulva, including varix, haematoma, oedema and gangrene; the treatment is the same as for the same disease in other parts. (4) The vulva is likely to be affected by a number of cutaneous affections, the most important being erythema, eczema, herpes, lichen, tubercle, elephantiasis, vulvitis pruriginosa, syphilis and kraurosis. These affections present the same characters as in other parts of the body. Kraurosis vulvae, first described by Lawson Tait in 1875, is an atrophic change accompanied by pain and a yellowish discharge; the cause is unknown. Pruritis vulvae is due to parasites, or to irritating discharges, as leucorrhoea, and is frequent in diabetic subjects. The hymen may be occasionally imperforate and require incision. Cysts and painful carunculae may occur on the clitoris. Any part of the vulva may be the seat of new growths, simple or malignant.

Diseases of the Vagina.—(1) Malformations. The vagina may be absent in whole or in part or may present a septum. Stenosis of the vagina may be a barrier to menstruation. (2) Displacements of the vagina; (a) cystocele, which is a hernia of the bladder into the vagina; (b) rectocele, a hernia of the rectum into the vagina. The cause of these conditions is relaxation of the tissues due to parturition. The palliative treatment consists in keeping up the parts by the insertion of a pessary; when this fails operative interference is called for. (3) Fistulae may form between the vagina and bladder or vagina and rectum; they are generally caused by injuries during parturition or the late stages of carcinoma. Persistent fistulae require operative treatment. The vagina normally secretes a thin opalescent acid fluid derived from the lymph serum and the shedding of squamous epithelium. This fluid normally contains the vagina bacillus. In pathological conditions of the vagina this secretion undergoes changes. For practical purposes three varieties of vaginitis may be described: (a) simple catarrhal vaginitis is due to the same causes as simple vulvitis, and occasionally in children is important from a medico-legal aspect when it is complicated by vulvitis. The symptoms are heat and discomfort with copious mucopurulent discharge. The only treatment required is rest, with vaginal douches of warm unirritating lotions such as boracic acid or subacetate of lead. (b) Gonorrhoeal vaginitis is most common in adults. The patient complains of pain and burning, pain on passing water and discharge which is generally green or yellow. The results of untreated gonorrhoeal vaginitis are serious and far-reaching. The disease may spread up the genital passages, causing endometritis, salpingitis and septic peritonitis, or may extend into the bladder, causing cystitis. Strict rest should be enjoined, douches of carbolic acid (1 in 40) or of perchloride of mercury (1 in 2000) should be ordered morning and evening, the vagina being packed with tampons of iodoform gauze. Saline purgatives and alkaline diuretics should be given, (c) Chronic vaginitis (leucorrhoea or “the whites”) may follow acute conditions and persist indefinitely. The vagina is rarely the seat of tumours, but cysts are common.

Diseases of the Uterus.—The uterus undergoes important changes during life, chiefly at puberty and at the menopause. At puberty it assumes the pear shape characteristic of the mature uterus. At the menopause it shares in the general atrophy of the reproductive organs. It is subject to various disorders and misplacements. (a) Displacements of the Uterus.—The normal position of the uterus, when the bladder is empty, is that of anteversion. We have therefore to consider the following conditions as pathological: anteflexion, retroflexion, retroversion, inversion, prolapse and procidentia. Slight anteflexion or bending forwards is normal; when exaggerated it gives rise to dysmenorrhoea, sterility and reflex nervous phenomena. This condition is usually congenital and is often associated with under-development of the uterus, from which the sterility results. The treatment is by dilatation of the canal or by a plastic operation. Retroflexion is a bending over of the uterus backwards, and occurs as a complication of retroversion (or displacement backwards). The causes are (1) any cause tending to make the fundus or upper part of the uterus extra heavy, such as tumours or congestion, (2) loss of tone of the uterine walls, (3) adhesions formed after cellulitis, (4) violent muscular efforts, (5) weakening of the uterine supports from parturition. The symptoms are dysmenorrhoea, pain on defaecation and constipation from the pressure of the fundus on the rectum; the patient is often sterile. The treatment is the replacing of the uterus in position, where it can be kept by the insertion of a pessary; failing this, operative treatment may be required. Retroversion when pathological is rarer than retroflexion. It may be the result of injury or is associated with pregnancy or a fibroid. The symptoms are those of retroflexion with feeling of pain and weight in the pelvis and desire to micturate followed by retention of urine due to the pressure of the cervix against the base of the bladder. The uterus must be skilfully replaced in position; when pessaries fail to keep it there the operation of hysteropexy gives excellent results.

Inversion occurs when the uterus is turned inside out. It is only possible when the cavity is dilated, either after pregnancy or by a polypus. The greater number of cases follow delivery and are acute. Chronic inversions are generally due to the weight of a polypus. The symptoms are menorrhagia, metrorrhagia and bladder troubles; on examination a tumour-like mass occupies the vagina. Reduction of the condition is often difficult, particularly when the condition has lasted for a long time. The tumour which has caused the inversion must be excised. Prolapse and procidentia are different degrees of the same variety of displacement. When the uterus lies in the vagina it is spoken of as prolapse, when it protrudes through the vulva it is procidentia. The causes are directly due to increased intra-abdominal pressure, increased weight of the uterus by fibroids, violent straining, chronic cough and weakening of the supporting structures of the pelvic floor, such as laceration of the vagina and perinaeum. Traction on the uterus from below (as a cervical tumour) may be a cause; advanced age, laborious occupations and frequent pregnancies are indirect causes. The symptoms are a “bearing down” feeling, pain and fatigue in walking, trouble with micturition and defaecation. The condition is generally obvious on examination. As a rule the uterus is easy to replace in position. A rubber ring pessary will often serve to keep it there. If the perinaeum is very much torn it may be necessary to repair it. Various operations for retaining the uterus in position are described. (b) Enlargements of the Uterus (hypertrophy or hyperplasia). This condition may sometimes involve the uterus as a whole or may be most marked in the body or in the cervix. It follows chronic congestion or inflammatory 766 prolapse, or any condition interfering with the circulation. The symptoms comprise local discomfort and sometimes dysmenorrhoea, leucorrhoea or menorrhagia. When the elongation occurs in the cervical portion the only possible treatment is amputation of the cervix. Atrophy of the uterus is normal after the menopause. It may follow the removal of the tubes and ovaries. Some constitutional diseases produce the same result, as tuberculosis, chlorosis, chronic morphinism and certain diseases of the central nervous system.

(c) Injuries and Diseases resultant from Pregnancy.—The most frequent of these injuries is laceration of the cervix uteri, which is frequent in precipitate labour. Once the cervix is torn the raw surfaces become covered by granulations and later by cicatricial tissue, but as a rule they do not unite. The torn lips may become unhealthy, and the congestion and oedema spread to the body of the uterus. A lacerated cervix does not usually give rise to symptoms; these depend on the accompanying endometritis, and include leucorrhoea, aching and a feeling of weight. Lacerations are to be felt digitally. As lacerations predispose to abortion the operation of trachelorraphy or repair of the cervix is indicated. Perforation of the uterus may occur from the use of the sound in diseased conditions of the uterine walls. Superinvolution means premature atrophy following parturition. Subinvolution is a condition in which the uterus fails to return to its normal size and remains enlarged. Retention of the products of conception may cause irregular haemorrhages and may lead to a diagnosis of tumour. The uterus should be carefully explored.

(d) Inflammations Acute and Chronic.—The mucous membrane lining the cervical canal and body of the uterus is called the endometrium. Acute inflammation or endometritis may attack it. The chief causes are sepsis following labour or abortion, extension of a gonorrhoeal vaginitis, or gangrene or infection of a uterine myoma. The puerperal endometritis following labour is an avoidable disease due to lack of scrupulous aseptic precautions.

Gonorrhoeal endometritis is an acute form associated with copious purulent discharge and well-marked constitutional disturbance. The temperature ranges from 99° to 105° F., associated with pelvic pain, and rigors are not uncommon. The tendency is to recovery with more or less protracted convalescence. The most serious complications are extension of the disease and later sterility. Rest in bed and intrauterine irrigation, followed by the introduction of iodoform pencils into the uterine cavity, should be resorted to, while pain is relieved by hot fomentations and sitz baths. Chronic endometritis may be the sequela of the acute form, or may be septic in origin, or the result of chronic congestion, acute retroflection or subinvolution following delivery or abortion. The varieties are glandular, interstitial, haemorrhagic and senile. The symptoms are disturbance of the menstrual function, headache, pain and pelvic discomfort, and more or less profuse thick leucorrhoeal discharge. The treatment consists in attention to the general health, with suitable laxatives and local injections, and in obstinate cases curettage is the most effectual measure. The disease is frequently associated with adenomatous disease of the cervix, formerly called erosion. In this disease there is a new formation of glandular elements, which enlarge and multiply, forming a soft velvety areola dotted with pink spots. This was formerly erroneously termed ulceration. The cause is unknown. It occurs in virgins as well as in mothers, but it often accompanies lacerations of the cervix. The symptoms are indefinite pain and leucorrhoea. The condition is visible on inspection with a speculum. The treatment is swabbing with iodized phenol or curettage. The body of the uterus may also be the seat of adenomatous disease. Tuberculosis may attack the uterus; this usually forms part of a general tuberculosis.

(e) New Growths in the Uterus.—The uterus is the most common seat of new growths. From the researches of von Gurlt, compiled from the Vienna Hospital Reports, embracing 15,880 cases of tumour, females exceed males in the proportion of seven to three, and of this large majority uterine growths account for 25%. When we consider its periodic monthly engorgements and the alternate hypertrophy and involution it undergoes in connexion with pregnancy, we can anticipate the special proneness of the uterus to new growths. Tumours of the uterus are divided into benign and malignant. The benign tumours known as fibroids or myomata are very common. They are stated by Bayle to occur in 20% of women over 35 years of age, but happily in a great number of cases they are small and give rise to no symptoms. They are definitely associated with the period of sexual activity and occur more frequently in married women than in single, in the proportion of two to one (Winckel). It is doubtful if they ever originate after the menopause. Indeed if uncomplicated by changes in them they share in the general atrophy of the sexual organs which then takes place. They are divided according to their position in the tissues into intramural, subserous and submucous (the last when it has a pedicle forms a polypus), or as to the part of the uterus in which they develop into fibroids of the cervix and fibroids of the body. Intramural and submucous fibroids give rise to haemorrhage. The menses may be so increased that the patient is scarcely ever free from haemorrhage. The pressure of the growth may cause dysmenorrhoea, or pressure on the bladder and rectum may cause dysuria, retention or rectal tenesmus. The uterus may be displaced by the weight of the tumour. Secondary changes take place in fibroids, such as mucous degeneration, fatty metamorphosis, calcification, septic infection (sloughing fibroid) and malignant (sarcomatous) degeneration.

The modes in which fibroids imperil life are haemorrhage (the commonest of all), septic infection, which is one of the most dangerous, impaction when it fits the true pelvis so tightly that the tumour cannot rise, twisting of the pedicle by rotation, leading to sloughing and intestinal and urinary obstruction. When fibroids are complicated by pregnancy, impaction and consequent abortion may take place, or a cervical myoma may offer a mechanical obstacle to delivery or lead to serious post partem haemorrhage. In the treatment of fibroids various drugs (ergot, hamamelis, hydrastis canadensis) may be tried to control the haemorrhage, and repose and the injection of hot water (120° F.) are sometimes successful, together with electrical treatment. Surgical measures are needed, however, in severe recurrent haemorrhage, intestinal obstruction, sloughing and the co-existence of pregnancy. An endeavour must be made if possible to enucleate the fibroid, or hysterectomy (removal of the uterus) may be required. The operation of removal of the ovaries to precipitate the menopause has fallen into disuse.

(f) Malignant Disease of the Uterus.—The varieties of malignant disease met with in the uterus are sarcoma, carcinoma and chorion-epithelioma malignum. Sarcomata may occur in the body and in the neck. They occur at an earlier age than carcinomata. Marked enlargement and haemorrhage are the symptoms. The differential diagnosis is microscopic. Extirpation of the uterus is the only chance of prolonging life. The age at which women are most subject to carcinoma (cancer) of the uterus is towards the decline of sexual life. Of 3385 collected cases of cancer of the uterus 1169 occurred between 40 and 50, and 856 between 50 and 60. In contradistinction to fibroid tumours it frequently arises after the menopause. It may be divided into cancer of the body and cancer of the neck (cervix). Cancer of the neck of the uterus is almost exclusively confined to women who have been pregnant (Bland-Sutton). Predisposing causes may be injuries during delivery. The symptoms which induce women to seek medical aid are haemorrhage, foetid discharge, and later pain and cachexia. An unfortunate belief amongst the public that the menopause is associated with irregular bleeding and offensive discharges has prevented many women from seeking medical advice until too late. It cannot be too widely understood that cancer of the cervix is in its early stages a purely local disease, and if removed in this stage usually results in cure. So important is the recognition of this fact in the saving of human life that at the meeting of the British Medical Association in April 1909 the council issued for publication a special appeal to medical practitioners, midwives and nurses, and directed it to be published in British and colonial medical and nursing journals. It will be useful to quote here a part of the appeal directed to midwives and nurses: “Cancer may occur at any age and in a woman who looks quite well, and who may have no pain, no wasting, no foul discharge and no profuse bleeding. To wait for pain, wasting, foul discharge or profuse bleeding is to throw away the chance of successful treatment. The early symptoms of cancer of the womb are:—(1) bleeding which occurs after the change of life, (2) bleeding after sexual intercourse or after a vaginal douche, (3) bleeding, slight or abundant, even in young women, if occurring between the usual monthly periods, and especially when accompanied by a bad-smelling or watery blood-tinged discharge, (4) thin watery discharge occurring at any age.” On examination the cervix presents certain characteristic signs, though these may be modified according to the variety of cancer present. Hard nodules or definite loss of substance, extreme friability and bleeding after slight manipulation, are suspicious. Epithelial cancer of the cervix may assume a proliferating ulcerative type, forming the well-known “cauliflower” excrescence. The treatment of cancer of the cervix is free removal at the earliest possible moment. Cancer of the body of the uterus is rare before the 45th year. It is most frequent at or subsequent to the menopause. The majority of the patients are nulliparae (Bland-Sutton). The signs are fitful haemorrhages after the menopause, followed by profuse and offensive discharges. The uterus on examination often feels enlarged. The diagnosis being made, hysterectomy (removal of the uterus) is the only treatment. Cancer of the body of the uterus may complicate fibroids. Chorion-epithelioma malignum (deciduoma) was first described in 1889 by Sänger and Pfeiffer. It is a malignant disease presenting microscopic characters resembling decidual tissue. It occurs in connexion with recent pregnancy, and particularly with the variety of abortion termed hydatid mole. In many cases it destroys life with a rapidity unequalled by any other kind of growth. It quickly ulcerates and infiltrates the uterine tissues, forming metastatic growths in the lung and vagina. Clinically it is recognized by the occurrence after pregnancy of violent haemorrhages, progressive cachexia and fever with rigors. Recent suggestions have been made as to chorion-epithelioma being the result of pathological changes in the lutein tissue of the ovary. The growth is usually primary in the uterus, but may be so in the Fallopian tubes and in the vagina. A few cases have been recorded unconnected with pregnancy. The virulence of chorion-epithelioma varies, but in the present state of our knowledge immediate removal of the primary growth along with the affected organ is the only treatment.

Diseases of the Fallopian Tubes.—The Fallopian tubes or oviducts 767 are liable to inflammatory affections, tuberculosis, sarcomata, cancer, chorion-epithelioma and tubal pregnancy. Salpingitis (inflammation of the oviducts) is nearly always secondary to septic infection of the genital tract. The chief causes are septic endometritis following labour or abortion, gangrene of a myoma, gonorrhoea, tuberculosis and cancer of the uterus; it sometimes follows the specific fevers. When the pus escapes from the tubes into the coelom it sets up pelvic peritonitis. When the inflammation is adjacent to the ostium it leads to the matting together of the tubal fimbriae and glues them to an adjacent organ. This seals the ostium. The occluded tube may now have an accumulation of pus in it (pyosalpinx). When in consequence of the sealing of the ostium the tube becomes distended with serous fluid it is termed hydrosalpinx. Haematosalpinx is a term applied to the non-gravid tube distended with blood; later the tubes may become sclerosed. Acute septic salpingitis is ushered in by a rigor, the temperature rising to 103°, 104° F., with severe pain and constitutional disturbance. The symptoms may become merged in those of general peritonitis. In chronic disease there is a history of puerperal trouble followed by sterility, with excessive and painful menstruation. Acute salpingitis requires absolute rest, opium suppositories and hot fomentations. With urgent symptoms removal of the inflamed adnexa must be resorted to. Chronic salpingitis often renders a woman an invalid. Permanent relief can only be afforded by surgical intervention. Tuberculous salpingitis is usually secondary to other tuberculous infections. The Fallopian tubes may be the seat of malignant disease. This is rarely primary. By far the most important of the conditions of the Fallopian tubes is tubal pregnancy (or ectopic gestation). It is now known that fertilization of the human ovum by the spermatozoon may take place even when the ovum is in its follicle in the ovary, for oosperms have been found in the ovary and Fallopian tubes as well as in the uterus. Belief in ovarian pregnancy is of old standing, and had been regarded as possible but unproved, no case of an early embryo in its membranes in the sac of an ovary being forthcoming, until the remarkable case published by Dr Catherine van Tussenboek of Amsterdam in 1899 (Bland-Sutton). Tubal pregnancy is most frequent in the left tube; it sometimes complicates uterine pregnancy; rarely both tubes are pregnant. When the oosperm lodges in the ampulla or isthmus it is called tubal gestation; when it is retained in the portion traversing the uterine wall it is called tubo-uterine gestation. Wherever the fertilized ovum remains and implants its villi the tube becomes turgid and swollen, and the abdominal ostium gradually closes. The ovum in this situation is liable to apoplexy, forming tubal mole. When the abdominal ostium remains pervious the ovum may escape into the coelomic cavity (tubal abortion); death from shock and haemorrhage into the abdominal cavity may result. When neither of these occurrences has taken place the ovum continues to grow inside the tube, the rupture of the distended tube usually taking place between the sixth and the tenth week. The rupture of the tube may be intraperitoneal or extraperitoneal. The danger is death from haemorrhage occurring during the rupture, or adhesions may form, the retained blood forming a haematocele. The ovum may be destroyed or may continue to develop. In rare cases rupture may not occur, the tube bulging into the peritoneal cavity; and the foetus may break through the membranes and lie free among the intestines, where it may die, becoming encysted or calcified. The tubal placenta possesses foetal structures, the true decidua forming in the uterus. The signs suggestive of tubal pregnancy before rupture are missed periods, pelvic pains and the presence of an enlarged tube. When rupture takes place it is attended in both varieties with sudden and severe pain and more or less marked collapse, and a tumour may or may not be felt according to the situation of the rupture. There is a general “feeling of something having given way.” If diagnosed before rupture, the sac must be removed by abdominal section. In intraperitoneal rupture immediate operation affords the only chance of saving life. In extraperitoneal rupture the foetus may occasionally remain alive until full term and be rescued by abdominal section, if the condition is recognized, or a false labour may take place, accompanied by death of the foetus.

Diseases of the Ovaries and Parovarium.—The ovaries undergo striking changes at puberty, and again at the menopause, after which there is a gradual shrinkage. One or both may be absent or malformed, or they are subject to displacements, being either undescended, contained in a hernia or prolapsed. Either of these conditions, if a source of pain, may necessitate their removal. The ovary is also subject to haemorrhage or apoplexy. Acute inflammations (oöphorites) are constantly associated with salpingitis or other septic conditions of the genital tract or with an attack of mumps. The relation of oöphoritis to mumps is at present unknown. Acute oöphoritis may culminate in abscess but more usually adhesions are formed. The surgical treatment is that of pyosalpinx. Chronic inflammation may follow acute or be consequent on pelvic cellulitis. Its constant features are more or less pain followed by sterility. The ovary may be the seat of tuberculosis, which is generally secondary to other lesions. Suppuration and abscess of the ovary also occur. Perioöphoritis, or chronic inflammation in the neighbourhood, may also involve the gland. The cause of cirrhosis of the ovaries is unknown, though it may be associated with cirrhotic liver. The change is met with in women between 20 and 40 years of age, the ovaries being in a shrunken, hard, wrinkled condition. Under ovarian neuralgia are grouped indefinite painful symptoms occurring frequently in neurotic and alcoholic subjects, and often worse during menstruation. The treatment, whether local or operative, is usually unsatisfactory. The ovary is frequently the seat of tumours, dermoids and cysts. Cysts may be simple, unilocular or multilocular, and may attain an enormous size. The largest on record was removed by Dr Elizabeth Reifsnyder of Shanghai, and contained 100 litres of fluid, and the patient recovered. The operation is termed ovariotomy. Dermoid cysts containing skin, bones, teeth and hair, are of frequent growth in the ovary, and have attained the weight of from 20 to 40 kilogrammes. In one case a girl weighed 27 kilogrammes and her tumour 44 kilogrammes (Keen). Papillomatous cysts also occur in the ovary. Parovarian and Gärtnerian cysts are found, and adenomata form 20% of all ovarian cysts. Occasionally the tunic of peritoneum surrounding the ovary becomes distended with serous fluid. This is termed ovarian hydrocele. Ovarian fibroids occur, and malignant disease (sarcoma and carcinoma) is fairly frequent, sarcoma being the most usual ovarian tumour occurring before puberty. Carcinoma of the ovary is rarely primary, but it is a common situation for secondary cancer to that of the breast, gall-bladder or gastro-intestinal tract. The treatment of all rapidly-growing tumours of the ovary is removal.

Diseases of the Pelvic Peritoneum and Connective Tissue.—Women are excessively liable to peritoneal infections. (1) Septic infection often follows acute salpingitis and may give rise to pelvic peritonitis (perimetritis), which may be adhesive, serous or purulent. It may follow the rupture of ovarian or dermoid cysts, rupture of the uterus, extra uterine pregnancy or extension from pyosalpinx. The symptoms are severe pain, fever, 103° F. and higher, marked constitutional disturbances, vomiting, restlessness, even delirium. The abdomen is fixed and tympanitic. Its results are the formation of adhesions causing abnormal positions of the organs, or chronic peritonitis may follow. The treatment is rest in bed, opium, hot stupes to the abdomen and quinine. (2) Epithelial infections take place in the peritoneum in connexion with other malignant growths. (3) Hydroperitoneum, a collection of free fluid in the abdominal cavity, may be due to tumours of the abdominal viscera or to tuberculosis of the peritoneum. (4) Pelvic cellulitis (parametritis) signifies the inflammation of the connective tissue between the folds of the broad ligament (mesometrium). The general causes are septic changes following abortion, delivery at term (especially instrumental delivery), following operations on the uterus or salpingitis. The symptoms are chill followed by severe intrapelvic pain and tension, fever 100° to 102° F. There may be nausea and vomiting, diarrhoea, rectal tenseness and dysuria. If consequent on parturition the lochia cease or become offensive. On examination there is tenderness and swelling in one flank and the uterus becomes fixed and immovable in the exudate as if embedded in plaster of Paris. The illness may go to resolution if treated by rest, opium, hot stupes or icebags and glycerine tampons, or may go on to suppuration forming pelvic abscess, which signifies a collection of pus between the layers of the broad ligament. The pus in a pelvic abscess may point and escape through the walls of the vagina, rectum or bladder. It occasionally points in the groin. If the pus can be localized an incision should be made and the abscess drained. The tumours which arise in the broad ligament are haematocele, solid tumours (as myomata, lipomata and sarcomata), and echinnococcus colonies (hydatids).

Bibliography.—Albutt, Playfair and Eden, System of Gynaecology (1906); McNaughton Jones, Manual of Diseases of Women (1904); Bland-Sutton and Giles, Diseases of Women (1906); C. Lockyer, “Lutein Cysts in association with Chorio-Epithelioma,” Journal of Obstetrics and Gynaecology (January, 1905); W. Stewart McKay, History of Ancient Gynaecology; Hart and Barbour, Diseases of Women; Howard Kelly, Operative Gynaecology.

The earliest edition of his collected poetical works is by Dugonics (Pressburg and Pest, 1796); the best modern selection is that of Toldy, entitled Gyöngyösi István válogatott poétai munkái (Select poetical works of Stephen Gyöngyösi, 2 vols., 1864-1865).

Theory of the Symmetrical Top.

1. The physical constants of a given symmetrical top, expressed in C.G.S. units, which are employed in the subsequent formulae, are denoted by M, h, C and A. M is the weight in grammes (g) as given by the number of gramme weights which equilibrate the top when weighed in a balance; h is the distance OG in centimetres (cm.) between G the centre of gravity and O the point of support, and Mh may be called the preponderance in g.-cm.; Mh and M can be measured by a spring balance holding up in a horizontal position the axis OC in fig. 8 suspended at O. Then gMh (dyne-cm. or ergs) is the moment of gravity about O when the axis OG is horizontal, gMh sin θ being the moment when the axis OG makes an angle θ with the vertical, and g = 981 (cm./s2) on the average; C is the moment of inertia of the top about OG, and A about any axis through O at right angles to OG, both measured in g-cm.2.

To measure A experimentally, swing the top freely about O in small plane oscillation, and determine the length, l cm., of the equivalent simple pendulum; then

l = A/Mh, A = Mhl.

Next make the top, or this simple pendulum, perform small conical revolutions, nearly coincident with the downward vertical position of equilibrium, and measure n, the mean angular velocity of the conical pendulum in radians / second; and T its period in seconds; then

4π2/T2 = n2 = g/l = gMh/A;

and f = n/2π is the number of revolutions per second, called the frequency, T = 2π/n is the period of a revolution, in seconds.

2. In the popular explanation of the steady movement of the top at a constant inclination to the vertical, depending on the composition of angular velocity, such as given in Perry’s Spinning Tops, or Worthington’s Dynamics of Rotation, Steady motion of the top. it is asserted that the moment of gravity is always generating an angular velocity about an axis OB perpendicular to the vertical plane COC′ through the axis of the top OC′; and this angular velocity, compounded with the resultant angular velocity about an axis OI, nearly coincident with OC′, causes the axes OI and OC′ to keep taking up a new position by moving at right angles to the plane COC′, at a constant precessional angular velocity, say μ rad./sec., round the vertical OC (fig. 4).

If, however, the axis OC′ is prevented from taking up this precessional velocity, the top at once falls down; thence all the ingenious attempts—for instance, in the swinging cabin of the Bessemer ship—to utilise the gyroscope as a mechanical directive agency have always resulted in failure (Engineer, October 1874), unless restricted to actuate a light relay, which guides the mechanism, as in steering a torpedo.

An experimental verification can be carried out with the gyroscope in fig. 1; so long as the vertical spindle is free to rotate in its socket, the rapidly rotating wheel will resist the impulse of tapping on the gimbal by moving to one side; but when the pinch screw prevents the rotation of the vertical spindle in the massive pedestal, this resistance to the tapping at once disappears, provided the friction of the table prevents the movement of the pedestal; and if the wheel has any preponderance, it falls down.

Familiar instances of the same principles are observable in the movement of a hoop, or in the steering of a bicycle; it is essential that the handle of the bicycle should be free to rotate to secure the stability of the movement.

The bicycle wheel, employed as a spinning top, in fig. 4, can also be held by the stalk, and will thus, when rotated rapidly, convey a distinct muscular impression of resistance to change of direction, if brandished.

3. A demonstration, depending on the elementary principles of dynamics, of the exact conditions required for the Elementary demonstration of the condition of steady motion. axis OC′ of a spinning top to spin steadily at a constant inclination θ to the vertical OC, is given here before proceeding to the more complicated question of the general motion, when θ, the inclination of the axis, is varying by nutation.

It is a fundamental principle in dynamics that if OH is a vector representing to scale the angular momentum of a system, and if Oh is the vector representing the axis of the impressed couple or torque, then OH will vary so that the velocity of H is represented to scale by the impressed couple Oh, and if the top is moving freely about O, Oh is at right angles to the vertical plane COC′, and

Oh = gMh sin θ.

771

In the case of the steady motion of the top, the vector OH lies in the vertical plane COC′, in OK suppose (fig. 4), and has a component OC = G about the vertical and a component OC′ = G′, suppose, about the axis OC; and G′ = CR, if R denotes the angular velocity of the top with which it is spun about OC′.

If μ denotes the constant precessional angular velocity of the vertical plane COC′ the components of angular velocity and momentum about OA are μ sin θ and Aμ sin θ, OA being perpendicular to OC′ in the plane COC′; so that the vector OK has the components

OC′ = G′, and C′K = Aμ sin θ,

and the horizontal component

CK = OC′ sin θ − C′K cos θ
= G′ sin θ − Aμ sin θ cos θ.

The velocity of K being equal to the impressed couple Oh,

gMh sin θ = μ·CK = sin θ (G′μ − Aμ2 cos θ),

and dropping the factor sin θ,

Aμ2 cos θ − G′μ + gMh = 0, or Aμ2 cos θ − CRμ + An2 = 0,

the condition for steady motion.

Solving this as a quadratic in μ, the roots μ1, μ2 are given by

and the minimum value of G′ = CR for real values of μ is given by

for a smaller value of R the top cannot spin steadily at the inclination θ to the upward vertical.

Interpreted geometrically in fig. 4

μ = gMh sin θ/CK = An2/KN, and μ = C′K/A sin θ = KM/A,

KM·KN = A2 n2,

so that K lies on a hyperbola with OC, OC′ as asymptotes.

4. Suppose the top or gyroscope, instead of moving freely about the point O, is held in a ring or frame which is compelled Constrained motion of the gyroscope. to rotate about the vertical axis OC with constant angular velocity μ; then if N denotes the couple of reaction of the frame keeping the top from falling, acting in the plane COC’, equation (4) § 3 becomes modified into

gMh sin θ − N = μ·CK = sin θ G′μ − Aμ2 cos θ,

N = sin θ (Aμ2 cos θ − G′μ + gMh)
= A sin θ cos θ (μ − μ1) (μ − μ2);

and hence, as μ increases through μ2 and μ1, the sign of N can be determined, positive or negative, according as the tendency of the axis is to fall or rise.

When G′ = CR is large, μ2 is large, and

μ1 ≈ gMh/G′ = An2/CR,

the same for all inclinations, and this is the precession observed in the spinning top and centrifugal machine of fig. 10 This is true accurately when the axis OC′ is horizontal, and then it agrees with the result of the popular explanation of § 2.

If the axis of the top OC′ is pointing upward, the precession is in the same direction as the rotation, and an increase of μ from μ1 makes N negative, and the top rises; conversely a decrease of the procession μ causes the axis to fall (Perry, Spinning Tops, p. 48).

If the axis points downward, as in the centrifugal machine with upper support, the precession is in the opposite direction to the rotation, and to make the axis approach the vertical position the precession must be reduced.

This is effected automatically in the Weston centrifugal machine (fig. 10) used for the separation of water and Centrifugal machine. molasses, by the friction of the indiarubber cushions above the support; or else the spindle is produced downwards below the drum a short distance, and turns in a hole in a weight resting on the bottom of the case, which weight is dragged round until the spindle is upright; this second arrangement is more effective when a liquid is treated in the drum, and wave action is set up (The Centrifugal Machine, C. A. Matthey).

Similar considerations apply to the stability of the whirling bowl in a cream-separating machine.

We can write equation (1)

N = An2 sin θ − μ·CK = (A2n2 − KM·KN) sin θ/A,

so that N is negative or positive, and the axis tends to rise or fall according as K moves to the inside or outside of the hyperbola of free motion. Thus a tap on the axis tending to hurry the precession is equivalent to an impulse couple giving an increase to C′K, and will make K move to the interior of the hyperbola and cause the axis to rise; the steering of a bicycle may be explained in this way; but K1 will move to the exterior of the hyperbola, and so the axis will fall in this second more violent motion.

Friction on the point of the top may be supposed to act like a tap in the direction opposite to the precession; and so the axis of a top spun violently rises at first and up to the vertical position, but falls away again as the motion dies out. Friction considered as acting in retarding the rotation may be compared to an impulse couple tending to reduce OC′, and so make K and K1 both move to the exterior of the hyperbola, and the axis falls in both cases. The axis may rise or fall according to the direction of the frictional couple, depending on the shape of the point; an analytical treatment of the varying motion is very intractable; a memoir by E. G. Gallop may be consulted in the Trans. Camb. Phil. Soc., 1903.

The earth behaves in precession like a large spinning top, of which the axis describes a circle round the pole of the ecliptic of mean angular radius θ, about 23½°, in a period of 26,000 years, so that R/μ = 26000 × 365; and the mean couple producing precession is

CRμ sin θ = CR2 sin 23½° /26000 × 365,

one 12 millionth part of ½CR2, the rotation energy of the earth.

5. If the preponderance is absent, by making the C·G coincide with O, and if Aμ is insensible compared with G′,

N = −G′μ sin θ,

the formula which suffices to explain most gyroscopic action.

Thus a carriage running round a curve experiences, in consequence Gyroscopic action of railway wheels. of the rotation of the wheels, an increase of pressure Z on the outer track, and a diminution Z on the inner, giving a couple, if a is the gauge,

Za = G′μ,

tending to help the centrifugal force to upset the train; and if c is the radius of the curve, b of the wheels, C their moment of inertia, and v the velocity of the train,

μ = v/c, G′ = Cv/b,

Z = Cv2/abc (dynes),

so that Z is the fraction C/Mab of the centrifugal force Mv2/c, or the fraction C/Mh of its transference of weight, with h the height of the centre of gravity of the carriage above the road. A Brennan carriage on a monorail would lean over to the inside of the curve at an angle α, given by

tan α = G′μ/gMh = G′v/gMhc.

The gyroscopic action of a dynamo, turbine, and other rotating machinery on a steamer, paddle or screw, due to its rolling and pitching, can be evaluated in a similar elementary manner (Worthington, Dynamics of Rotation), and Schlick’s gyroscopic apparatus is intended to mitigate the oscillation.

6. If the axis OC in fig. 4 is inclined at an angle α to the vertical, the equation (2) § 4 becomes

N = sin θ (Aμ2 cos θ − G′μ) + gMh sin (α − θ).

Suppose, for instance, that OC is parallel to the earth’s axis, and that the frame is fixed in the meridian; then α is the co-latitude, and μ is the angular velocity of the earth, the square of which may be neglected; so that, putting N = 0, α − θ = E,

gMh sin E − G′μ sin (α − E) = 0,

This is the theory of Gilbert’s barogyroscope, described in Appell’s Mécanique rationnelle, ii. 387: it consists essentially of a rapidly The barogyroscope. rotated fly-wheel, mounted on knife-edges by an axis perpendicular to its axis of rotation and pointing east and west; spun with considerable angular momentum G′, and provided with a slight preponderance Mh, it should tilt to an angle E with the vertical, and thus demonstrate experimentally the rotation of the earth.

In Foucault’s gyroscope (Comptes rendus, 1852; Perry, p. 105) Foucault’s gyroscope. the preponderance is made zero, and the axis points to the pole, when free to move in the meridian.

Generally, if constrained to move in any other plane, the axis seeks the position nearest to the polar axis, like a dipping needle with respect to the magnetic pole. (A gyrostatic working model of the magnetic compass, by Sir W. Thomson. British Association Report, Montreal, 1884. A. S. Chessin, St Louis Academy of Science, January 1902.)

A spinning top with a polished upper plane surface will provide an artificial horizon at sea, when the real horizon is obscured. The first instrument of this kind was constructed by Serson, and is described in the Gentleman’s Magazine, Gyroscopic horizon. vol. xxiv., 1754; also by Segner in his Specimen theoriae turbinum (Halae, 1755). The inventor was sent to sea by the Admiralty to test his instrument, but he was lost in the wreck of the “Victory,” 1744. A copy of the Serson top, from the royal collection, is now in the Museum of King’s College, London. Troughton’s Nautical Top (1819) is intended for the same purpose.

The instrument is in favour with French navigators, perfected by 772 Admiral Fleuriais (fig. 9); but it must be noticed that the horizon given by the top is inclined to the true horizon at the angle E given by equation (3) above; and if μ1 is the precessional angular velocity as given by (3) § 4, and T = 2π/μ, its period in seconds,

if E is expressed in minutes, taking μ = 2π/86400; thus making the true latitude E nautical miles to the south of that given by the top (Revue maritime, 1890; Comptes rendus, 1896).

This can be seen by elementary consideration of the theory above, for the velocity of the vector OC′ of the top due to the rotation of the earth is

μ·OC′ cos lat = gMh sin E = μ1·OC′ sin E,

in which 8π can be replaced by 25, in practice; so that the Fleuriais gyroscopic horizon is an illustration of the influence of the rotation of the earth and of the need for its allowance.

7. In the ordinary treatment of the general theory of the gyroscope, the motion is referred to two sets of rectangular axes; the Euler’s coordinate angles. one Ox, Oy, Oz fixed in space, with Oz vertically upward and the other OX, OY, OZ fixed in the rotating wheel with OZ in the axis of figure OC.

The relative position of the two sets of axes is given by means of Euler’s unsymmetrical angles θ, φ, ψ, such that the successive turning of the axes Ox, Oy, Oz through the angles (i.) ψ about Oz, (ii.) θ about OE, (iii.) φ about OZ, brings them into coincidence with OX, OY, OZ, as shown in fig. 11, representing the concave side of a spherical surface.

The component angular velocities about OD, OE, OZ are

ψ sin θ, θ, φ + ψ cos θ;

so that, denoting the components about OX, OY, OZ by P, Q, R,

Consider, for instance, the motion of a fly-wheel of preponderance Mh, and equatoreal moment of inertia A, of which the axis OC is held in a light ring ZCX at a constant angle γ with OZ, while OZ is held by another ring zZ, which constrains it to move round the vertical Oz at a constant inclination θ with constant angular velocity μ, so that

θ = 0, ψ = μ;

P = μ sin θ sin φ, Q = μ sin θ cos φ, R = φ + μ cos θ.

With CXF a quadrant, the components of angular velocity and momentum about OF, OY, are

P cos γ − R sin γ, Q, and A (P cos γ − R sin γ), AQ,

so that, denoting the components of angular momentum of the fly-wheel about OC, OX, OY, OZ by K or G′, h1, h2, h3,

h1 = A (P cos γ − R sin γ) cos γ + K sin γ,

h2 = AQ,

h3 = −A (P cos γ − R sin γ) sin γ + K cos γ;

and the dynamical equation

with K constant, and with preponderance downward

N = gMh cos zY sin γ = gMh sin γ sin θ cos φ,

reduces to

+ Aμ2 cos γ sin θ cos θ cos φ − (Kμ + gMh) sin θ cos φ = 0.

The position of relative equilibrium is given by

For small values of μ the equation becomes

so that φ = ½π gives the position of stable equilibrium, and the period of a small oscillation is 2π √{A sin γ/(Kμ + gMh) sin θ}.

In the general case, denoting the periods of vibration about φ = ½π, −½π, and the sidelong position of equilibrium by 2π/(n1, n2, or n3), we shall find

n3 = n1 n2/μ sin θ.

The first integral of (11) gives

− Aμ2 cos γ sin θ cos θ sin φ + (Kμ + gMh) sin θ sin φ − H = 0,

and putting tan (¼π + ½φ) = z, this reduces to

where Z is a quadratic in z2, so that z is a Jacobian elliptic function of t, and we have

tan (¼π + ½φ) = C (tn, dn, nc, or cn) nt,

according as the ring ZC performs complete revolutions, or oscillates about a sidelong position of equilibrium, or oscillates about the stable position of equilibrium φ = ±½π.

Suppose Oz is parallel to the earth’s axis, and μ is the diurnal rotation, the square of which may be neglected, then if Gilbert’s barogyroscope of § 6 has the knife-edges turned in azimuth to make an angle β with E. and W., so that OZ lies in the horizon at an angle E·β·N., we must put γ = ½π, cos θ = sin α sin β; and putting φ = ½π − δ + E, where δ denotes the angle between Zz and the vertical plane Zζ through the zenith ζ,

sin θ cos δ = cos α, sin θ sin δ = sin α cos β;

so that equations (9) and (10) for relative equilibrium reduce to

gMh sin E = KQ = Kμ sin θ cos φ = Kμ sin θ sin (δ − E),

and will change (3) § 6 into

a multiplication of (3) § 6 by cos β (Gilbert, Comptes rendus, 1882).

Changing the sign of K or h and E and denoting the revolutions/second of the gyroscope wheel by F, then in the preceding notation, T denoting the period of vibration as a simple pendulum,

so that the gyroscope would reverse if it were possible to make F cos α > 86400 A/T2C (Föppl, Münch. Ber, 1904).

A gyroscopic pendulum is made by the addition to it of a fly-wheel, balanced and mounted, as in Gilbert’s barogyroscope, in a ring movable about an axis fixed in the pendulum, in the vertical plane of motion.

As the pendulum falls away to an angle θ with the upward vertical, and the axis of the fly-wheel makes an angle φ with the vertical plane of motion, the three components of angular momentum are

h1 = K cos φ, h2 = Aθ + K sin φ, h3 = Aφ,

where h3 is the component about the axis of the ring and K of the fly-wheel about its axis; and if L, M′, N denote the components of the couple of reaction of the ring, L may be ignored, while N is zero, with P = 0, Q = θ, R = 0, so that

M′ = h2 = Aθ + Kφ cos φ,

0 = h3 − h1θ = Aφ − Kθ cos φ.

For the motion of the pendulum, including the fly-wheel,

MK2θ = gMH sin θ − M′ = gMH sin θ − Aθ − Kφ cos φ.

If θ and φ remain small,

Aφ = Kθ, Aφ = K(θ − α),

(MK2 + A) θ + (K2/A) (θ − α) − gMHθ = 0;

so that the upright position will be stable if K2 > gMHA, or the rotation energy of the wheel greater than ½A/C times the energy acquired by the pendulum in falling between the vertical and horizontal position; and the vibration will synchronize with a simple pendulum of length

(MK2 + A) / [(K2/gA) − MH].

This gyroscopic pendulum may be supposed to represent a ship among waves, or a carriage on a monorail, and so affords an explanation of the gyroscopic action essential in the apparatus of Schlick and Brennan.

8. Careful scrutiny shows that the steady motion of a top is not steady absolutely; it reveals a small nutation General motion of the top. superposed, so that a complete investigation requires a return to the equations of unsteady motion, and for the small oscillation to consider them in a penultimate form.

In the general motion of the top the vector OH of resultant angular momentum is no longer compelled to lie in the vertical plane COC′ (fig. 4), but since the axis Oh of the gravity couple is always horizontal, H will describe a curve in a fixed horizontal plane through C. The vector OC′ of angular momentum about the axis will be constant in length, but vary in direction; and OK will be the component angular momentum in the vertical plane COC′, if the planes through C and C′ perpendicular to the lines OC and OC′ intersect in the line KH; and if KH is the component angular momentum perpendicular to the plane COC′, the resultant angular momentum OH has the three components OC′, C′K, KH, represented in Euler’s angles by

KH = A dθ/dt, C′K = A sin θd ψ/dt, OC′ = G′.

Drawing KM vertical and KN parallel to OC′, then

KM = A dψ/dt, KN = CR − A cos θ dψ/dt = (C − A) R + A dφ/dt

so that in the spherical top, with C = A, KN = A dφ/dt.

773

The velocity of H is in the direction KH perpendicular to the plane COC′, and equal to gMh sin θ or An2 sin θ, so that if a point in the axis OC′ at a distance An2 from O is projected on the horizontal plane through C in the point P on CK, the curve described by P, turned forwards through a right angle, will be the hodograph of H; this is expressed by

where ρeῶi is the vector CH; and so the curve described by P and the motion of the axis of the top is derived from the curve described by H by a differentiation.

Resolving the velocity of H in the direction CH,

d·CH/dt = An2 sin θ sin KCH = An2 sin θ KH/CH,

d·½CH2/dt = A2n2sin θ dθ/dt.

and integrating

½CH2 = A2n2 (E − cos θ),

½OH2 = A2n2 (F − cos θ),

½C′H2 = A2n2 (D − cos θ),

where D, E, F are constants, connected by

F = E + G2/2A2n2 = D + G′2/2A2n2.

Then

KH2 = OH2 − OK2,

OK2 sin2 θ = CC′2 = G2 − 2GG′ cos θ + G′2,

A2 sin2 θ (dθ/dt)2 = 2A2n2 (F − cos θ) sin2 θ − G2 + 2GG′ cos θ − G′2;

and putting cos θ = z,

Denoting the roots of Z = 0 by z1, z2, z3, we shall have them arranged in the order

z1 > 1 > z2 > z > z3 > −1.

(dz/dt)2 = 2n2 (z1 − z) (z2 − z) (z − z3).

nt = ∫ zz3 dz/ √(2Z),

an elliptic integral of the first kind, which with

can be expressed, when normalized by the factor √(z1 − z3)/2, by the inverse elliptic function in the form

z − z3 = (z2 − z3) sn2mt, z2 − z = (z2 − z3) cn2mt, z1 − z = (z1 − z3) dn2mt.

z = z2sn2mt + z3cn2mt.

Interpreted dynamically, the axis of the top keeps time with the beats of a simple pendulum of length

L = l/½ (z1 − z3),

suspended from a point at a height ½ (z1 + z3)l above O, in such a manner that a point on the pendulum at a distance

½ (z1 − z3) l = l2/L

from the point of suspension moves so as to be always at the same level as the centre of oscillation of the top.

The polar co-ordinates of H are denoted by ρ, ῶ in the horizontal plane through C; and, resolving the velocity of H perpendicular to CH,

ρdῶ/dt = An2 sin θ cos KCH.

ρ2dῶ/dt = An2 sin θ·CK = An2 (G′ − G cos θ)

an elliptic integral, of the third kind, with pole at z = E; and then

ῶ − ψ = KCH = tan−1 KH/CH

which determines ψ.

Otherwise, from the geometry of fig. 4,

C′K sin θ = OC − OC′ cos θ,

A sin2 θ dψ/dt = G − G′ cos θ,

the sum of two elliptic integrals of the third kind, with pole at z = ±1; and the relation in (25) (26) shows the addition of these two integrals into a single integral, with pole at z = E.

The motion of a sphere, rolling and spinning in the interior of a spherical bowl, or on the top of a sphere, is found to be of the same character as the motion of the axis of a spinning top about a fixed point.

The curve described by H can be identified as a Poinsot herpolhode, that is, the curve traced out by rolling a quadric surface with centre fixed at O on the horizontal plane through C; and Darboux has shown also that a deformable hyperboloid made of the generating lines, with O and H at opposite ends of a diameter and one generator fixed in OC, can be moved so as to describe the curve H; the tangent plane of the hyperboloid at H being normal to the curve of H; and then the other generator through O will coincide in the movement with OC′, the axis of the top; thus the Poinsot herpolhode curve H is also the trace made by rolling a line of curvature on an ellipsoid confocal to the hyperboloid of one sheet, on the plane through C.

Kirchhoff’s Kinetic Analogue asserts also that the curve of H is the projection of a tortuous elastica, and that the spherical curve of C′ is a hodograph of the elastica described with constant velocity.

Writing the equation of the focal ellipse of the Darboux hyperboloid through H, enlarged to double scale so that O is the centre,

x2/α2 + y2/β2 + z2/O = 1,

with α2 + λ, β2 + λ, λ denoting the squares of the semiaxes of a confocal ellipsoid, and λ changed into μ and ν for a confocal hyperboloid of one sheet and of two sheets.

λ > 0 > μ > −β2 > ν > −α2,

then in the deformation of the hyperboloid, λ and ν remain constant at H; and utilizing the theorems of solid geometry on confocal quadrics, the magnitudes may be chosen so that

α2 + λ + β2 + μ + ν = OH2 = ½k2 (F − z) = ρ2 + OC2.

α2 + μ = ½k2 (z1 − z) = ρ2 − ρ12,

β2 + μ = ½k2 (z2 − z) = ρ2 − ρ22,

μ = ½k2 (z3 − z) = ρ2 − ρ32,

ρ12 < 0 < ρ22 < ρ2 < ρ32,

F = z1 + z2 + z3,

λ − 2μ + ν = k2z, λ − ν = k2,

with z = cos θ, θ denoting the angle between the generating lines through H; and with OC = δ, OC′ = δ′, the length k has been chosen so that in the preceding equations

δ/k = G/2An, δ′/k = G′/2An;

and δ, δ′, k may replace G, G′, 2An; then

while from (33-39)

which verifies that KH is the perpendicular from O on the tangent plane of the hyperboloid at H, and so proves Darboux’s theorem.

Planes through O perpendicular to the generating lines cut off a constant length HQ = δ, HQ′ = δ′, so the line of curvature described by H in the deformation of the hyperboloid, the intersection of the fixed confocal ellipsoid λ and hyperboloid of two sheets ν, rolls on a horizontal plane through C and at the same time on a plane through C′ perpendicular to OC′.

Produce the generating line HQ to meet the principal planes of the confocal system in V, T, P; these will also be fixed points on the generator; and putting

(HV, HT, HP,)/HQ = D/(A, B, C,),

then

Ax2 + By2 + Cz2 = Dδ2

is a quadric surface with the squares of the semiaxes given by HV·HQ, HT·HQ, HP·HQ, and with HQ the normal line at H, and so touching the horizontal plane through C; and the direction cosines of the normal being

x/HV, y/HT, z/HP,

A2x2 + B2y2 + C2z2 = D2δ2,

the line of curvature, called the polhode curve by Poinsot, being the intersection of the quadric surface (44) with the ellipsoid (46).

There is a second surface associated with (44), which rolls on the plane through C′, corresponding to the other generating line HQ′ through H, so that the same line of curvature rolls on two planes at a constant distance from O, δ and δ′; and the motion of the top is made up of the combination. This completes the statement of Jacobi’s theorem (Werke, ii. 480) that the motion of a top can be resolved into two movements of a body under no force.

Conversely, starting with Poinsot’s polhode and herpolhode given in (44) (46), the normal plane is drawn at H, cutting the principal axes of the rolling quadric in X, Y, Z; and then

α2 + μ = x·OX, β2 + μ = y·OY, μ = z·OZ,

this determines the deformable hyperboloid of which one generator through H is a normal to the plane through C; and the other generator is inclined at an angle θ, the inclination of the axis of the top, while the normal plane or the parallel plane through O revolves with angular velocity dψ/dt.

The curvature is useful in drawing a curve of H; the diameter of curvature D is given by

774

The curvature is zero and H passes through a point of inflexion when C′ comes into the horizontal plane through C; ψ will then be stationary and the curve described by C′ will be looped.

In a state of steady motion, z oscillates between two limits z2 and z3 which are close together; so putting z2 = z3 the coefficient of z in Z is

With z2 = z3, κ = 0, K = ½π; and the number of beats per second of the axis is

beating time with a pendulum of length

The wheel making R/2π revolutions per second,

from (8) (9) § 3; and the apsidal angle is

and the height of the equivalent conical pendulum λ is given by

if OR drawn at right angles to OK cuts KC′ in R, and RL is drawn horizontal to cut the vertical CO in L; thus if OC2 represents l to scale, then OL will represent λ.

9. The gyroscope motion in fig. 4 comes to a stop when the rim of the wheel touches the ground; and to realize the motion when the axis is inclined at a greater angle with the upward vertical, the stalk is pivoted in fig. 8 in a lug screwed to the axle of a bicycle hub, fastened vertically in a bracket bolted to a beam. The wheel can now be spun by hand, and projected in any manner so as to produce a desired gyroscopic motion, undulating, looped, or with cusps if the stalk of the wheel is dropped from rest.

As the principal part of the motion takes place now in the neighbourhood of the lowest position, it is convenient to measure the angle θ from the downward vertical, and to change the sign of z and G.

Equation (18) § 8 must be changed to

1 > z3 > z > z2 > −1, D, E > z1,

z1 + z2 + z3 = F = D − G′2 / 2A2n2 = E − G2 / 2A2n2,

and expressed by the inverse elliptic function

z = z2sn2mt + z3cn2mt, κ2 = (z3 − z2) / (z3 − z1).

Equation (25) and (29) § 8 is changed to

while ψ and ῶ change places in (26).

The Jacobian elliptic parameter of the third elliptic integral in (7) can be given by ν, where

where f is a real fraction,

with respect to the comodulus κ′.

Then, with z = E, and

2ZE = −{ (G′ − GE) / An}2,

if II denotes the apsidal angle of ῶ, and T the time of a single beat of the axle, up or down,

= ½πf + Kznf K′,

in accordance with the theory of the complete elliptic integral of the third kind.

Interpreted geometrically on the deformable hyperboloid, flattened in the plane of the focal ellipse, if OQ is the perpendicular from the centre on the tangent HP, AOQ = amfK′, and the eccentric angle of P, measured from the minor axis, is am(1 − f) K′, the eccentricity of the focal ellipse being the comodulus κ′.

A point L is taken in QP such that

QL/OA = znfK′,

QV, QT, QP = OA (zs, zc, zd) fK′;

and with

mT = K, m/n = √ (z3 − z1) /2 = OA/k,

By choosing for f a simple rational fraction, such as ½, 1⁄3, ¼, 1⁄5, ... an algebraical case of motion can be constructed (Annals of Mathematics, 1904).

Thus with G′ − GE = 0, we have E = z1 or z2, never z3; f = 0 or 1; and P is at A or B on the focal ellipse; and then

ῶ = −pt, p = G/2A,

sin θ exp (ψ + pt) i = i√ [(−z2 − z3) (z − z1)] + √ [(z3 − z) (z − z2)],

sin θ exp (ψ + pt)i = i√ [(−z1 − z3) (z − z2)] + √ [(z3 − z) (z − z1)],

Thus z2 = 0 in (22) makes G′ = 0; so that if the stalk is held out horizontally and projected with angular velocity 2p about the vertical axis OC without giving any spin to the wheel, the resulting motion of the stalk is like that of a spherical pendulum, and given by

= i sin α √ (sec α cos θ) + √ [(sec α + cos θ) (cos α − cos θ)],

if the axis falls in the lowest position to an angle α with the downward vertical.

With z3 = 0 in (21) and z2 = −cos β, and changing to the upward vertical measurement, the motion is given by

sin θ eψi = eint √ ½ cos β [√ (1 − cos β cos θ) + i√ (cos β cos θ − cos2 θ)],

and the axis rises from the horizontal position to a series of cusps; and the mean precessional motion is the same as in steady motion with the same rotation and the axis horizontal.

The special case of f = ½ may be stated here; it is found that

ρ2 = a2 (κ − x2),

L = ½ (1 − κ) + λp/n,

so that p = 0 and the motion is made algebraical by taking L = ½ (1 − κ).

The stereoscopic diagram of fig. 12 drawn by T. I. Dewar shows these curves for κ = 15⁄17, 3⁄5, and 1⁄3 (cusps).

10. So far the motion of the axis OC’ of the top has alone been considered; for the specification of any point of the body, Euler’s third angle φ must be introduced, representing the angular displacement of the wheel with respect to the stalk. This is given by

It will simplify the formulas by cancelling a secular term if we make C = A, and the top is then called a spherical top; OH becomes the axis of instantaneous angular velocity, as well as of resultant angular momentum.

When this secular term is restored in the general case, the axis OI of angular velocity is obtained by producing Q′H to I, making

775

and then the four vector components OC′, C′K, KH, HI give a resultant vector OI, representing the angular velocity ω, such that

OI/Q′I = ω/R.

The point I is then fixed on the generating line Q′H of the deformable hyperboloid, and the other generator through I will cut the fixed generator OC of the opposite system in a fixed point O′, such that IO′ is of constant length, and may be joined up by a link, which constrains I to move on a sphere.

In the spherical top then,

depending on the two elliptic integrals of the third kind, with pole at z = ±1; and measuring θ from the downward vertical, their elliptic parameters are:—

Then if v′ = K + (1 − f′)K′i is the parameter corresponding to z = D, we find

f = f2 − f1, f′ = f2 + f1,

v = v1 + v2, v′ = v1 − v2.

The most symmetrical treatment of the motion of any point fixed in the top will be found in Klein and Sommerfeld, Theorie des Kreisels, to which the reader is referred for details; four new functions, α, β, γ, δ, are introduced, defined in terms of Euler’s angles, θ, ψ, φ, by

α = cos ½θ exp ½ (φ + ψ) i,

β = i sin ½θ exp ½ (−φ + ψ) i,

γ = i sin ½θ exp ½ (φ − ψ) i,

δ = cos ½θ exp ½ (−φ − ψ) i.

Next Klein takes two functions or co-ordinates λ and Λ, defined by

and Λ the same function of X, Y, Z, so that λ, Λ play the part of stereographic representations of the same point (x, y, z) or (X, Y, Z) on a sphere of radius r, with respect to poles in which the sphere is intersected by Oz and OZ.

These new functions are shown to be connected by the bilinear relation

in accordance with the annexed scheme of transformation of co-ordinates—

where

ξ = x + yi,   η = −x + yi,   ζ = −z,
Ξ = X + Yi,   Η = −X + Yi,   Ζ = −Z;

and thus the motion in space of any point fixed in the body defined by Λ is determined completely by means of α, β, γ, δ; and in the case of the symmetrical top these functions are elliptic transcendants, to which Klein has given the name of multiplicative elliptic functions; and

αδ = cos2 ½θ,   βγ = −sin2 ½θ,
αδ − βγ = 1,   αδ + βγ = cos θ,
√ ( −4αβγδ) = sin θ;

while, for the motion of a point on the axis, putting Λ = 0, or ∞,

λ = β/δ = i tan ½θeψi, or λ = α/γ = −i cot ½θeψi,

and

αβ = ½i sin θeψi, αγ = ½i sin θeψi,

giving orthogonal projections on the planes GKH, CHK; and

the vectorial equation in the plane GKH of the herpolhode of H for a spherical top.

When f1 and f2 in (9) are rational fractions, these multiplicative elliptic functions can be replaced by algebraical functions, qualified by factors which are exponential functions of the time t; a series of quasi-algebraical cases of motion can thus be constructed, which become purely algebraical when the exponential factors are cancelled by a suitable arrangement of the constants.

Thus, for example, with f = 0, f′ = 1, f1 = ½, f2 = ½, as in (24) § 9, where P and P′ are at A and B on the focal ellipse, we have for the spherical top

(1 + cos θ) exp (φ + ψ − qt) i
= √ (sec β − cos θ) √ (cos β − cos θ) + i (√ sec β + √ cos β) √ cos θ,

(1 − cos θ) exp (φ − ψ − q′t) i
= √ (sec β − cos θ) √ (cos β − cos θ) + i (√sec β − √ cos β) √ cos θ,

q, q′ = n√ (2 sec β) ± n√ (2 cos β);

and thence α, β, γ, δ can be inferred.

The physical constants of a given symmetrical top have been denoted in § 1 by M, h, A, C, and l, n, T; to specify a given state of general motion we have G, G′ or CR, D, E, or F, which may be called the dynamical constants; or κ, v, w, v1, v2, or f, f′, f1, f2, the analytical constants; or the geometrical constants, such as α, β, δ, δ′, k of a given articulated hyperboloid.

There is thus a triply infinite series of a state of motion; the choice of a typical state can be made geometrically on the hyperboloid, flattened in the plane of the local ellipse, of which κ is the ratio of the semiaxes α and β, and am(1 − f) K′ is the eccentric angle from the minor axis of the point of contact P of the generator HQ, so that two analytical constants are settled thereby; and the point H may be taken arbitrarily on the tangent line PQ, and HQ′ is then the other tangent of the focal ellipse; in which case θ3 and θ2 are the angles between the tangents HQ, HQ′, and between the focal distances HS, HS′, and k2 will be HS·HS′, while HQ, HQ′ are δ, δ′. 776 As H is moved along the tangent line HQ, a series of states of motion can be determined, and drawn with accuracy.

11. Equation (5) § 3 with slight modification will serve with the same notation for the steady rolling motion at a constant inclination α to the vertical of a body of revolution, such as a disk, hoop, wheel, cask, wine-glass, plate, dish, bowl, spinning top, gyrostat, or bicycle, on a horizontal plane, or a surface of revolution, as a coin in a conical lamp-shade.

The point O is now the intersection of the axis GC′ with the vertical through the centre B of the horizontal circle described by the centre of gravity, and through the centre M of the horizontal circle described by P, the point of contact (fig. 13). Collected into a particle at G, the body swings round the vertical OB as a conical pendulum, of height AB or GL equal to g/μ2 = λ, and GA would be the direction of the thread, of tension gM(GA/GL) dynes. The reaction with the plane at P will be an equal parallel force; and its moment round G will provide the couple which causes the velocity of the vector of angular momentum appropriate to the steady motion; and this moment will be gM·Gm dyne-cm. or ergs, if the reaction at P cuts GB in m.

Draw GR perpendicular to GK to meet the horizontal AL in R, and draw RQC′K perpendicular to the axis Gz, and KC perpendicular to LG.

The velocity of the vector GK of angular momentum is μ times the horizontal component, and

horizontal component /Aμ sin α = KC/KC′,

so that

gM·Gm = Aμ2 sin α(KC/KC′),

The instantaneous axis of rotation of the case of a gyrostat would be OP; drawing GI parallel to OP, and KK′ parallel to OG, making tan K′GC′ = (A/C) tan IGC’1; then if GK represents the resultant angular momentum, K′K will represent the part of it due to the rotation of the fly-wheel. Thus in the figure for the body rolling as a solid, with the fly-wheel clamped, the points m and Q move to the other side of G. The gyrostat may be supposed swung round the vertical at the end of a thread PA′ fastened at A′ where Pm produced cuts the vertical AB, and again at the point where it crosses the axis GO. The discussion of the small oscillation superposed on the state of steady motion requisite for stability is given in the next paragraph.

12. In the theoretical discussion of the general motion General motion of a gyrostat rolling on a plane. of a gyrostat rolling on a horizontal plane the safe and shortest plan apparently is to write down the most general equations of motion, and afterwards to introduce any special condition.

Drawing through G the centre of gravity any three rectangular axes Gx, Gy, Gz, the notation employed is

The geometrical equations, expressing that the point of contact is at rest on the plane, are

u − ry + qz = 0,

v − pz + rx = 0,

w − qx + py = 0.

The dynamical equations are

du/dt − θ3v + θ2w = gα + X/M,

dv/dt − θ1w + θ2u = gβ + Y/M,

dw/dt − θ2u + θ1v = gγ + Z/M,

and

dh1/dt − θ3h2 + θ2h3 = yZ − zY,

dh2/dt − θ1h3 + θ3h1 = zX − xZ,

dh3/dt − θ2h1 + θ1h2 = xY − yX.

In the special case of the gyrostat where the surface is of revolution round Gz, and the body is kinetically symmetrical about Gz, we take Gy horizontal and Gzx through the point of contact so that y = 0; and denoting the angle between Gz and the downward vertical by θ (fig. 13)

α = sin θ,   β = 0,   γ = cos θ.

The components of angular momentum are

h1 = Ap,   h2 = Aq,   h3 = Cr + K,

where A, C denote the moment of inertia about Gx, Gz, and K is the angular momentum of a fly-wheel fixed in the interior with its axis parallel to Gz; K is taken as constant during the motion.

The axis Gz being fixed in the body,

θ1 = p,   θ2 = q = −dθ/dt,   θ3 = p cot θ.

With y = 0, (1), (2), (3) reduce to

u = −qz,   v = pz − rx,   w = qx;

and, denoting the radius of curvature of the meridian curve of the rolling surface by ρ,

so that

The dynamical equations (4)...(9) can now be reduced to

Eliminating Y between (19) and (23),

Eliminating Y between (19) and (21)

− pqz (x + z cot θ − ρ sin θ) + qrzρ cos θ = 0,

+ pz (x + z cot θ − ρ sin θ) + rzρ cos θ = 0.

In the special case of a gyrostat rolling on the sharp edge of a circle passing through G, z = 0, ρ = 0, (A) and (B) reduce to

a differential equation of a hypergeometric series, of the form of Legendre’s zonal harmonic of fractional order n, given by

n (n + 1) = CMx2 / A (Mx2 + C).

For a sharp point, x = 0, ρ = 0, and the previous equations are obtained of a spinning top.

The elimination of X and Z between (18) (20) (22), expressed symbolically as

(22) − z(18) + x(20) = 0,

gives

+ q2ρ (x cos θ − z sin θ) − prx (x + z cot θ) − g (x cos θ − z sin θ) = 0,

and this combined with (A) and (B) will lead to an equation the integral of which is the equation of energy.

13. The equations (A) (B) (C) are intractable in this general form; but the restricted case may be considered when the axis moves in steady motion at a constant inclination α to the vertical; and the stability is secured if a small nutation of the axis can be superposed.

It is convenient to put p = Ω sin θ, so that Ω is the angular velocity of the plane Gzx about the vertical; (A) (B) (C) become

777

− Ωx (x sin θ − 2z cos θ − ρ sin2 θ) + rxρ cos θ = 0,

+ Ωz sin θ (x − ρ sin θ) − rzρ cos θ = 0,

− Ωrx (x sin θ + z cos θ) − g (x cos θ − z sin θ) = 0.

The steady motion and nutation superposed may be expressed by

θ = α + L, sin θ = sin α + L cos α, cos θ = cos α − L sin α, Ω = μ + N, r = R + Q,

where L, N, Q are small terms, involving a factor enti, to express the periodic nature of the nutation; and then if a, c denote the mean value of x, z, at the point of contact

x = a + Lρ cos α, z = c − Lρ sin α,

x sin θ + z cos θ = a sin α + c cos α + L (a cos α − c sin α),

x cos θ − z sin θ = a cos α − c sin α − L (a sin α + c cos α − ρ).

Substituting these values in (C*) with dq/dt = −d2θ/dt2 = n2L, and ignoring products of the small terms, such as L2, LN, ...

which is equivalent to

+ μ2 ac sin2 α − μRa (a sin α + c cos α) − g (a cos α − c sin α) = 0,

the condition of steady motion; and

DL + EQ + FN = 0,

where

+ 2μac sin2 α − Ra (a sin α + c cos α).

With the same approximation (A*) and (B*) are equivalent to