METHOD OP TRANSFORMING AND RESOLVING EQUATIONS. 327 



which shall satisfy a given system of h^ rational and integral 

 and homogeneous equations of the first degree, 



A' = 0, A" = 0, . . . A(*'^ = ; (193.) 



Aj such equations of the second degree, 



B' = 0, B" = 0, . . B^-^) = ; (194.) 



^3 of the third degree, 



C = 0, C" = 0, . . C^-^') = ; . . . . . . (195.) 



and so on, as far as hf equations of the /th degree, 



1^ = 0, T" = 0,..TW = o, (196.) 



without being obliged in any part of the process, to introduce 

 any elevation of degree by elimination." Mr. Jerrard has not 

 published his solution of this very general problem, but he has 

 sufficiently suggested the method which he would employ, and 

 it is proper to discuss it briefly here, with reference to the ex- 

 tent of its application, and the circumstances under which it 

 fails ; not only on account of the importance of such discussion 

 in itself, but also because it is adapted to throw light on all the 

 questions already considered. 

 If we asume 



o, = a\ + a\, a^ = a\, + a%, • • • «^ = a'^ + a"^, . . (197.) 



that is, if we decompose each of the m disposable quantities 

 a^ into two parts, we may then accordingly decom- 



«,, «, 



15 ""gJ 



pose every one of the Aj proposed homogeneous functions of 

 those m quantities, which are of the first degree, namely, 



A',A", ..A«, ..a(*>); (198.) 



every one of the h^ proposed functions of the second degree, 



B',B",..bW, ..B(*=); (199.) 



every one of the hg functions of the third degree, 



c',c",..cw,..c(*3)j (200.) 



and so on, as far as all the first h^ — 1 functions of the tth de- 

 gree, 



T', T", ..tW ..T(*t-i). (201.) 



(the last function T^*') being reserved for another purpose, which 



