This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1872 Excerpt: ...of all the derivatives taken with respect to (f). Hence, if the number of the functions (/, ) is the same with that of the variables, 2 D.FDF, Dx%Fn = (--y+mnZ DfFDfxFx l, nFn, and "n--K ) Djli-Dfx Fi 2)/. /; 364. If the number of the functions (/) exceeds that of the variables and is p--1 instead of n--1, let (Fj) be ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1872 Excerpt: ...of all the derivatives taken with respect to (f). Hence, if the number of the functions (/, ) is the same with that of the variables, 2 D.FDF, Dx%Fn = (--y+mnZ DfFDfxFx l, nFn, and "n--K ) Djli-Dfx Fi 2)/. /; 364. If the number of the functions (/) exceeds that of the variables and is p--1 instead of n--1, let (Fj) be the form of (F() when the last p--n of the functions (/) are eliminated from it by means of the last p--n of the given equations. In this case it follows from the reasoning of 354 that + Dxfd, yfx DFnDfn Fn+l D/pFp = 2 DXFDFX D.jl 2 + J)fH+1Fn + 1Fn+2 D, fF 2 DfFIflFx D/rFf But the equation (1892G) is applicable to this case if (F) is changed to (F), and, therefore, the introduction of a common factor into the terms of (lSO) gives, by means of the preceding equations, SB _/ Z+DsFF, DnFnD/, +, FH+x Mn--K ) 2 J), FD/1Fl D/, FP 365. There are various interesting and instructive relations between the partial determinants of functions which have been developed by Jacobi, and which will be found useful in discussing the theory of differential equations. If the number of the functions (fi) as well as of the variables (a-, ) is increased to m--n--1, let If, then, from the function (/n + 1), all the variables x, zlf zn_x are eliminated, and the functions f, fx introduced in their places, and the function (/ +, ) thus transformed is denoted by (fl+i), the values of become 2Bif = 9t, d f1, .. ---1 - + lV n + l The determinant of the (m-f-1)2 functions (Bp) is, consequently, 2 + 03 iliY..'))" a")= 53 -+- + D, jDx /-i A. /i. m. is in (191u) the factor of A/.+i But the factor of this same quantity in.-j., is, by inspection, (_)-s +1/ +2/1 U-iJm = ...
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