METHOD OF TRANSFORMING AND RESOLVING EQUATIONS, 337 



SO that at last we are conducted to a condition which may be 

 thus denoted, and which contains the ultimate result of all the 

 restrictions on the number m, 



m--^^-Vl-"A,_2-"^V3-----^'~^^^2>^'"%» (254.) 

 that is, 



m>A^+^/i,_i + "A,_2 + "V3 + --+^'"^^'^2 + ^'~^^^i- (255.) 

 The 7iumber m, of quantities originally disposable, must there- 

 fore in general be at least equal to a certain minor limit, which 

 may be thus denoted, 



m (^1, /ia, /?8, ...hf)-h^ + 'hf_^ + "A^_2 + • • • 1 ^^^^ ^ 



+ «-.)„ + ('-.)^, ^ 1, I <"^-' 



in order that the method may succeed; and reciprocally, the 

 method will in general be successful when m equals or sicrpasses 

 this limit. 



[15.] To illustrate the foregoing general discussion, let us 

 suppose that 



t = 2; (257.) 



that is, let us propose to satisfy a system containing Aj equa- 

 tions of the first degree, 



A' = 0, . . A^») = 0, . . A^'^') =0, . . (193.) 

 and Ag equations of the second degree, 



B' = 0, . . B^^) = 0, . . B(*2) = 0, . . (194.) 



(but not containing any equations of higher degrees than the 

 second,) by a suitable selection of the m — 1 ratios of m (quan- 

 tities, 



«i. •••««, (192.) 



and without being obliged in any part of the process to intro- 

 duce any elevation of degree by elimination. Assuming, as 

 before, 



«i = a\ + a\, . . . a^ = <„ + «"^, . (197-) 



and employing the corresponding decompositions 



A' = A'^^^ + A'„^^,...a(*«) = A^;;') + A^^;), . . . (258.) 



and 



B' = B',,o+B\,, + B'o,„.... 



T3(/'2 - 1) _ -0(^2 - 1) , Ti^' - 1) , T,{h, - 1) r • • (259.) 



^ ~ -^2,0 "'' -^1,1 ^ 0,2 ' 



we shall be able to resolve the original problem, if we can re- 

 solve the system of the three following. ^ 

 VOL. v.— 1836. z 



