METHOD OP TRANSFORMING AND RESOLVING EQUATIONS. 321 



and thereby to reduce the general equation of the m"* degree, 



X = a7"' + Aa?'»-^ + 60?"*-^ + C ^'"-^ + D ^'"-^ 

 + E o?*"-^ + &c. = 0, 



(1.) 



to the form 



Y = ^'" + E'^'"-^ + &c. = 0. . . (15.) 



It is possible, of course, that this may not be precisely the same 

 as Mr. Jerrard's unpublished process, but it seems likely that 

 the one would not be found to differ from the other in any essen- 

 tial respect, notation being always excepted. It is, at least, a 

 process suggested by the published researches of that author, 

 and harmonizing with the discoveries which they contain. 



But by applying to this new process the spirit of the former 

 discussions, and putting, for abbreviation, 



A>sJ.^') + A" VO + A"'5o(^"') + M'.^o^''') -f ... 



a'/ \ 



m—l 



m— I 



+ M' 





y\ 



m—i 

 Pm—l, 



+ 



...(164.) 



N'*o('') + ... + Nvi^o(''')=yo, 



m—l 



■'Vr) + .. 



S's, 





m—l m—l ■» m—l, 



+ N'L('') + ... + N^i L^'"") + S'L(?') + ... 

 + gviiiL(?v.")^L, 



(165./ 



(166.) 



(167.) 



we may change the expression (149) to the form (100), through 

 the theorem and notation (44) ; and in order to avoid the case 

 of failure, in which the functions <p {x) and x (^) in (1^3) are 

 VOL. V. — 1836. Y 



