Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/40

30 INFINITESIMAL CALCULUS Multiplying by h, and operating on Loth sides with the symbol of differentiation D, we get But, by analogy from 53, we may write flU Dj/l-lJ .Dj T l72~~ 1.2.3.4 + . . . Hence 70 = - -( x + - </>"( B 7/ 2 " " - 1.2. . This result is due to Stirling, and has important applications. To complete this proof it is necessary to consider the question of the convergency or divergency of this series. On this investiga tion see Bertram!, Calcul liMgral, Art. 374. 93. Again, in the calculus of finite differences, if we consider the finite symbol of summation 2 as the inverse to that of finite differ ences A, we have A - 1 B 2 D 3 "1.2.3.4 , ii^L 1.2.3.4 C + /<p(x)dx - 30240 This theorem is due to Euler ; the foregoing demonstration was given by Gregory (Camb. Math. Journal, 1837). On the limits of the remainder after n terms in this series, see Boole s Finite Differences, pp. 91-93 ; also Mr J. W. L. Glaisher, in Quarterly Journal of Mathematics, 1872. In concluding this brief account of symbolic methods we may observe that the general principles of the theory of operations have been studied in a comprehensive manner by Grassmann, and by Ilankel, who applied them to the general theory of complex vari ables and of quaternions. See Grassmann s A usdchnungslehre (1862), and Hankel s Vorlcsungcn iiber die Complexen Zahlcn, 1867. The reader will find Grassmann s method fully discussed in Houel s Calcul Infinitesimal, vol. i. We add a few miscellaneous examples of these methods. (1) Prove the symbolic equation (2) Trove that (3) Prove the symbolic equation in finite differences (E-a)"X = a+*A"a-"X, where E is the symbol C D (Gregory, Camb. Math. Jour., 1837). (4) If IT and p be symbols of operation such that irp- pir=--p i , Trp 1 - p L ir = p. 2 , Trp.,- p. 2 K = p 3 . . prove the following symbolic equation See Donkin, Camb. and Dub. Math. Jour., 1850. (5) From the preceding the following symbolic equations can be readily deduced. (Donkin, ibid.) (3) Every differential equation of the form -(a + bx + ex- + . . .)D l + (a + l x + . . . )D"- l +. . .^; can be transformed into the shape f (xD)+f 1 (xT>)x+f. 2 (xD)^+ . . ,j:=X. (Boole.) (7) Apply the method of operations to the proof of HersclicTs nsion for F(c ( ) (Philosophical we make = in the equation expansion for F(c ( ) (Philosophical Transactions, 1813). If w we have /(<)=/(D)c oe , where D represents the symbol -- If now/(0 = F(e<), we get F(e ) = F(c B ) . c ( = F(l + A)c _ (Gregory, Camb. Math. Jour., 1838.) (8) Prove the equation &c. where D represents r -, and E represents dx Also (Bronwin, Camb. and Dub. Math. Jour., 1848.) (9) Prove the symbolic equation where x is to be regarded as a variable independent of the opera tion D, but which, after the operations, is to be replaced by x. (Crofton, Quar. Math. Jour., 1879; also Donkin, Camb. and Dub. Math. Jour., 1850.) Change of Independent Variable. 94. In the application of the calculus it is often necessary" to adopt in our equations new independent variables instead of those originally selected. Thus, suppose it be required to transform a function of y, x, c !l dy & c ., into a function of y, t, ^ , | , x being supposed a dx dx 2 at dV function of t. cPx Let the functions dt ~ at* then we have in all cases du_ 1 du dx x dt &c., be represented by x , x", &c., dy dx ~l_(li dx x dt J x dt x dt . d ~y _ j&y tPy_ d ( " dP dt} 1 d_ x dt x 1 dt also Again and so on for differentials of higher order. If y be taken as the independent variable, we have dt dt Hence "^ = 0, &c. 1 d?x dy_l^ cPy^ dx~dx dx*~ (dx* dy* dy) 31 1 I "5 5 / <V dx d*x The formulre for the change of the independent variable were given for the first time in the Traits dcs infinimcnt pctitcs of L H6j)it;il. The general theory of transformation was discussed at considerable extent by Euler in his Calc. Diff. In the case of two independent variables, suppose we are given dv Then dv de dv dx dv di/ dx dr dy dr dv dx dv dy dx dQ dy ~dG