554 Mr DE MORGAN, ON VARIOUS POINTS, &c. 



If we develope U in powers of w, x, y, we have for the coefficient of w m x n y p : (l . 2...»i) 

 (l.2...n) (1.2... p) the value (w = 0, x-=0, y = 0) of 



tfn + n+PfJ Jm + n+P-1 



or 



idU\ d".+»+*-i J W m X"Y p dU\ 



1 Vdy) ' ° r d*™*"-*"- 1 ( Z m+ " +p ~dz) ; 



dw m dx"df dw'"dafdf- 1 \dy t 



u, in the last, being derived from ssZ + ... = 0. This residual equation ought to be of finite 

 solution, though cases might possibly arise in which it would be desirable to apply the whole 



theorem to the residual equation itself, and to obtain a sufficient number of cases of W m 



{dU:d%) expanded in powers of gf. 



The following is a real extension. Let u be defined by the equation 



F(u, wW + xX + ..., w 1 W 1 + w l X 1 + ..., w, d W 2 + x 2 X 2 +..., ) = 0. 



The first theorems in (1), with the rule of intrusion and extrusion, remain true for 

 all variables which are in the same set : the second theorems for all variables whatsoever. 

 Hence U may be expanded, from a residual equation, in powers of any of the variables. 

 For example, suppose 



u = F (<3? + f <pu, y + riyu, % + £\|/w). 



The general term of the development of U has for the coefficient of £V"£"-J- ( 2 • 3... I) 

 &c. either 



u^A&^^m or ^*r xtt> ^)' &c - : 



in which u = F (x, y, %). The extension given by Laplace (Berlin Memoirs, 1777, p. 116 ; 

 Lacroix, Vol. I. p. 282), which exhibits the case of expansion of a function of u, «, &c, * 

 where u = F (x + XQ, v = G (y + Yrj), &c, in which X, Y, &c. are functions of u, v, &c, 

 may be both simplified and extended by the above mode of treatment. 



A. DE MORGAN. 



December 20, 1854. 



