r 



by means of SymmetmaFuneUom^B -t) /i 1/ 3SX 



assigned to it, there will result . ^n i;»^ t:- -r^}h v^r -rrfiifj^^^n 

 %iy2y3.-yn=S.(«P+)SQ+7R+ .. +XLr; 



a theorem, the truth of which is admitted at the present day*. 



3. Thus we perceive that in the problem of the transforma- 

 tion of equations, however vast it may be, we shall merely have 

 tQi^o|^i4^r functions of the class rr ^ ., * /^ 



,3JqmBX3 T 6 . (aPH-/3Q + 7ll+ . . +\L)", if^rfi Ito ptoirob- 



'*'^fKe^^rBo^#yc$i t|ave of iiblii my Mathematicall^earcferSii^ 

 twenty years ago was nearly as follows : — 



Let us, in the first place, suppose that 



R=0, ..L=0. ~* 

 The expression for 2y^ y-i'-Vn will thus become '"'^ f"=? "' .litxidw illO'ii 



This function is, we perceive, of n dimensions relatively to P and Q ^:<af§| 

 see also that an ath power of each of the n quantities Xi, w^, . . Xn is suc- 

 cessively joined to P, and a jSth power to Q ; hence we are led to conclude 

 that ^ i 



SDflo i& ed IIlw =vo®«"P"+vi(&«'^-^^ P" ^Q ni «-«\ lo iflob 

 -hv2(5«"~^^2p«-2Q2_,_ .. ^vn^^^'Q"; ro 5d o;t naaa 



Vq, Vi, Vg, . . Vn being certain constant but unknown quantities. 

 Now in order to determine these, let 



we shall then have^'^^>^^^^-^ va ^^dJ a^lJ aw |i 



And since, m general, .-ib-ooa'fi tuff 



2(P+Q) «.,«.,.. ^n=(P^^i^^^ h^miohamt 

 =(P+Q^'50^ . <. ^: -5- 

 there will result, on making the requisite substitution, 



(PH-Q)«=VoP^+ViP«->Q+V2P"-'Q2+ . . ^-v„Q^ 



to(w— 1) S^Liiad St 

 Thus vq, Vj, V2, ..vm are respectively equal to I, w, a .2,^^ iiiijt»^^^ 



coefficients of the development of (P+Q)". .^/^ '^ 



If, therefore, we introduce the equation of definition 



1.2 



,ilfmn oor + !i(^ ^^^-^^2 p«-2Q2+ . . +@^»Q^ '^ ^'^ 



we shall finally obtam 



;;;!^;;.2(P<+Qa.f)(Pa.J+Q^f)..(P<+Q^f)=©.(«P+ ' '^^ 



And from this we can ascend without difficulty to the general fofm^.^. 

 which R, . . L are undetermined and arbitrary. . aint't ^ 



