V A R V A R goal. Now the radical changes sign as s goes roundoff, and again as it goes round b ; hence it returns to a with the same sign. Thus Oa is travelled in the / two directions with the root having the same sign ; hence 7(0 ) and 7(aO) cancel. By de creasing indefinitely the radii of the circles, the circular integrals vanish ; ^ / and, as the root is negative along a/3 and positive along /3a, the value of A - B reduces to the negative double value of the in tegral from a to (3, that is, from a to b for vanishing radii ; thus dz /n - - 2 V1 - /- - - 2 V ^ ie rec tilinear inte grals required are now only between the limits 0, 1, and I/A 1 , so that they do not cross any singular point. In case k is a positive proper fraction we have in Jacobi s notation A = 2 A", A C 47i", A - B = 1iK ; therefore all f* dz values of the elliptic integral / j= , what- ./ v(l - -)(! - k-z 2 ) ever be the path taken between the limits and z, are included in the two formulae 4/iT+ ImiK + w and knK + IrniK + 2A - w. Hence the inverse function z = sin amw is doubly periodic for 4AT and 2t 7t", thus sin am(knK+ ImiK + w) = sinaimv sin am(2(2)i + 1)71"+ ImiK - ?/>) = sina? ?>-. This last shows that sin am w takes the same value for two different values of w within the same period. In the plane of w mea sure a series of par allels (see fig. 8) to the axis of ordinates at distances 47iT, and to the prime axis at ime v in lea- to ites B Tz Pa - ind > at
- ...
S di- <b c P "" & ft
- O^
w goes 4 A 1C
u the plane will be di vided into rect- angles such that in each s mamw through all its values and has equal values at corresponding points of differ ent rectangles : thus, whatever it is at P for w = u + iv, it is the same again at P v P 2 , and 7*., ; at Q, where w = 2K -u + i(2K - v), it has the same value ; at TT!, where w = 2K + u + iv, and at S, where w = kK u + i(2K - v), it has the equal and opposite negative value. The value of sin am iv vanishes at the points marked in fig. 8 by circles and is infinite at those denoted by crosses. (B. w.) VARIATION AND SELECTION IT is not proposed in the present article to trace the successive steps by which the general doctrine of the origin of species by descent with modification has come to gain acceptance among naturalists (see EVOLUTION). The present problem is concerned solely with the deter minant factors of evolution, with searching out the mechanism of the evolutionary process, and of discerning if possible such order as may lie under the apparent flux of change. Yet a brief retrospect of the essential views of successive evolutionists is still necessary, but only in so far as these have been concerned, not simply with the empirical evidence for descent with modification, but with the deeper attempt to explain the rationale of this process, to show how it is that modifications come to arise, persist, and increase at all. In other words, just as there are these two essential aspects of the doctrine of evolution, which must on pain of immediate confusion be clearly kept apart, so it is necessary to distinguish between the corre sponding sides of its historical development, and, passing beyond the simple hypothesis and evidences of descent, consider the attempts to explain the actual process of modification. Theories of Variation. If we pass over the speculations of the earliest evolu tionists De Maillet, Maupertuis, Robinet, Bonnet, &c. as too vague for rapid summary, and note that Linnaeus was not quite a consistent creationist, since he admitted that many species may be simply fertile hybrids, the his tory of definite speculation as to the factors of variation ButTon. may be fairly said to begin with Buffon, Avhose distinct though covert suggestion of the doctrine of descent with modification was supported by emphatic insistence upon the importance of external conditions. He endeavoured to show how "natures, instincts, and most inward quali ties" are modified through bodily habits, how new functions seem to arise in response to new conditions, and how changes in climate, food, and other conditions of life bring about direct modifications in organisms exposed to them. Kant. Kant, on the other hand, viewed the evolution of species as related essentially to the mechanical laws of the organ ism itself, although in his latest writings he allowed for the influence of geographical distribution, food, &c., noted the importance of selection in artificial breeding, and even hinted at the notion of struggle for existence, which was soon afterwards more clearly emphasized by Herder (Ideen Herder. z. Phil. d. Geschichte, 1790) in a striking passage, which may be briefly quoted in free translation: "All is in struggle, each one for himself. . . . Space is too small for the number. . . . Each genus looks after itself as if it were the only one. ... In this way the whole was pre served. . . . Thus were forces cradled, and limbs counted, and tendencies determined, and the earth came to bear what it could." Erasmus Darwin (Zoonomia, 1794) believed that the Erasmus organism has the faculty of improving by its own inherent Darwin, activity, that it has the power of attaining new parts attended with new propensities. Yet the strongest and most active animals are those which propagate and hand on improvements. Transformations, too, may be produced in part by the exertions of the organism in consequence of its desires for food, security, and reproduction. Changed conditions, such as climate, have an indirect influence in changing desires, and thus actions, and so finally struc ture ; and they may also operate directly. Treviranus Trevir- (1802-1831) assumed an indefinite variability of the organ- anus. ism, with considerable power of adaptation to surroundings, and even anticipated much recent speculation in his sug gestion of a possible factor of modification in the union of sexual elements. The well-known theory of Lamarck Lamarck (1801-1809) laid special emphasis on function and en vironment ; for, though the sense of need in association with suitable environment calls out a succession of efforts, and so originates incipient structural modifications, it is to increased functioning that the increase of these modifi cations must be ascribed, while similarly disuse explains degeneration. Changed conditions produce new wants, nutritive and reproductive ; hence changes in climate, or the like, change the organism by changing its habits. Rapid increase is checked by other organisms : the strongest
and best-armed for attack devour the weaker, and the less