Page:Encyclopædia Britannica, Ninth Edition, v. 24.djvu/101

Rh V A R V A R 85 and propose an interpretation in essential respects directly converse, without denying the minor agency of use and dis use, environment, &c. Each of the greater steps of progress is definitely associated with an increased measure of sub ordination of individual competition to reproductive or .social ends, and of interspecific competition to co-operative adaptation. The ideal of evolution is thus an Eden ; and, although competition can never be wholly eliminated, and progress must thus be asymptotic, it is much for our pure natural history to see no longer struggle, but love, as &quot;creation s final law.&quot; While ceasing to speak of inde finite variation, we may of course still conveniently retain the rest of the established phraseology, and continue to speak of &quot;natural selection &quot; and of &quot;survival of the fit test,&quot; always provided that, in passing from the explana tion of the distributional survival of individuals or species in contest within a given area to the interpretation of the main line of their morphological and physiological progress, we make the transition from the self-regarding to the other-regarding (in ethical language, from the egoistic to the altruistic) sense of these terms which has above been outlined. 1 It would be premature to enter upon the extended or deductive application of these considerations, since, pend ing their acceptance, the preceding statement of the received doctrine of natural selection from spontaneous variations, with all its logical consequences (pp. 81-82), remains valid. Conversely, it is of course obvious that their adoption would involve the extensive modification of the received doctrine, as well as the complementing of new constructive attempts by a re-examination of earlier views of both the process and the philosophy of evolution. 2 (p. GE.) VARIATIONS, CALCULUS OF. 1. It has been observed (MAXIMA AND MINIMA, vol. xv. p. 643) that the origin of the calculus of variations may be traced to John Ber noulli s celebrated problem, published in 1696 in the Acta Eruditorum of Leipsic, under the following form, Datis in piano verticali duobus punctis A et J3, assiynare mobili J/ viam AMB per qiiam gravitate sua descendens, et moveri inci])iens a puncto A, brevissimo tempore perveniat ad punctum B. This problem introduced considerations en tirely different from those hitherto involved in the discus sion of curves, for in its treatment it is necessary to conceive a curve as changing its form in a continuous manner, that is, as undergoing what is styled deformation. This change of form can be treated analytically as follows. Suppose y =/(#) to represent the equation of a curve, and let us write y =f(^;) + a^(x) (1), where a is an infinitesimal quantity, and &amp;lt;/ (&amp;lt; ) any function of x subject only to the condition of being finite for all values of x within the limits of the problem. Then equation (1) represents a new curve indefinitely close to the curve y =f(x) ; and by varying the form of }/(x) we may regard (1) as representative of any continuous curve indefinitely near to the original curve. 2. Again al/(x) is the difference of the y ordinates of the two curves for the same value of x ; this indefinitely small difference is called the variation of y, and is denoted by 8t/. It may be regarded as the change in y arising solely from a change in the relation connecting y with jr, while ,c itself remains unaltered. In general, if y be a function of any number of independent variables x v j 2,. . . jr n, then 8y represents any indefinitely small change in y arising solely from a change in the form of the function, while x v x. 2, &c., are unchanged. Thus the variable y may receive two essentially distinct kinds of increment,- one arising from a change in one or more of the variables, the other arising solely from a change in the relation which connects y with these variables. The former increments are those Contemplated and treated of in the ordinary calculus ; the latter are those principally considered in the calculus of variations. We shall follow Strauch, Jellett, Moigno, and the prin cipal modern writers on the subject, by restricting, in general, the symbol 8 to the latter species of increment. In many problems both kinds of increment have place simultaneously. Thus, if y=f(j v #&amp;lt;&amp;gt;, * 3 ), and if Ay denote the total increment of y, we have and so on in all cases. j ^ 3. It is readily seen that 8 7 =-r-8i/, and in general d n u d 1 that 8 ~ = -, 8y. The last equation may be written dx n dx n where D stands for the symbol of differentiation, - ax 4. We shall adopt Xewton s notation and write y for dy d 2 // d 1 // d 2 //

dx~ , y for 00 for, and proceed to consider the ~ variation of the general expression V=f(x, y, y, y . . . y (n) ), in which the form of the function /is given, while that of y in terms of x is indeterminate. Here, considering x as unchanged, we have _dV dV dV ,) Now let /- =P, - df -P v . . . - - - = /&amp;gt;, then we have 5 V= Poy + PJ}5i/ + P-.D dy + .. . + P,Jf8y (4). 5. Next let us consider the variation of the definite integral U= I Vdx, where V does not contain either of / ^ o the limits j&quot; and x^. Here, when the limits are unchanged, pj __ we have 8 U I 8 Vdx, and, when the limits undergo change, ./ r,i 6. We shall here bring in a new symbol, called the sign of substitution, which was introduced by Sarrus. 3 Thus, if F be any function of .r, y, &c., the result of substituting a-j for j- in F the other variables being unchanged is / Xl represented by / F in the form of notation adopted by / /.Tj Moigno and Lindelof. 4 The difference between / F said /F is represented by / F. Employing this notation, (5) is written 5 U= I 5 1 dx + / Vdx (6). ./ J o u 7. We now proceed to transform the last equation, commencing ,-ith the case in which Tis a function of .&amp;gt;, y, and y solely. Vdx, with Here 8U= Xext let V be a function of x, y, y, and i/, then

1 GecUles, &quot;On Variation in Plants,&quot; &quot;On the Origin of Thorns ami Prickles,&quot; and &quot; On the Origin of Evergreens,&quot; in Trans. Hot. Soc. Edhi., 1886, 1887, 1888. - Geddes, &quot; A Restatement of the Theory of Organic Evolution,&quot; Hoy. Soc. Edin., 1888. 3 Recherches si&amp;gt;r le Calcul des Variations, 1848. 4 C alc. des Var., 18(31.