METHOD OP TRANSFORMING AND RESOLVING EQUATIONS, 339 



auxiliary system of equations, (261) and (262), into a system 

 which may be thus denoted. 





(266.) 



and which contains h^ + hc^ — 1 equations of the first degree, 

 and Ag — 1 equations of the second degree, between the m — 1 

 new combinations, or new auxiliary quantities following, 



b, = a",^^a"^,...b^_, = a\_,-<L;zla"^; (267-) 



so that the solution of the second auxiliary problem will give, 

 in general, 



*i=0, ...Z»^_i = 0, (231.) 



and therefore will give, for the m auxiliary quantities (205), a 

 system of ratios coincident with the ratios (219), 



-^ = -^, ^ = ^y_l, 268.) 



unless 



m - 1 7 Ai + 2 (^2 - 1) (269.) 



When, therefore, this last condition is not satisfied, the two first 

 auxiliary problems will conduct, in general, to a system of de- 

 termined ratios for the m original quantities (192), namely 



——,... j , . . . ^^IS.} 



^m a'm ^m ^ m 



and unless these happen to satisfy the equation of the third 

 auxiliary problem, namely 



BW = 0, (263.) 



which had not been employed in determining them, we shall 

 fall back on the excluded case, or case of failure, (216). But, 

 even when the condition (269) is satisfiied, and when, therefore, 

 the auxiliary equations are theoretically capable of conducting 

 to ratios which shall satisfy the equations originally proposed, 

 it will still be necessary, in general, to decompose each of the 

 two first auxiliary systems of equations into others, in order to 

 comply with the enunciation of the original problem, which re- 

 quires that we should avoid all raising of degree by elimination, 



z 2 



