METHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 333 



which are rational and integral and homogeneous with respect 

 to the m — 1 quantities (226), and are independent of the mul- 

 tiplier a. Unless, then, the number of the equations of this 

 transformed system (230), which is the same as the number of 

 equations in the second auxiliary problem before proposed, be 

 less than the number m — 1 of the new auxiliary quantities 

 (226), we shall have, in general, null values for all those quanti- 

 ties, that is, we shall have 



Z», = 0,A2 = 0, ...Z.^_i = 0; (23L) 



and therefore we shall be conducted, by (222), to expressions of 

 the forms (217), which will in general lead, as has been already 

 shown, to the case of failure (216). We have therefore a new 

 condition of inequality, which the number m must satisfy, in 

 order to the general success of the method, namely the follow- 



m - 1 > A\ + A'a + A'a + . . . + A^; • • • • (232.) 

 in which, h\, h'^, h'^, " • h'^. denote respectively the numbers of 

 the equations of the first, second, third, . . . and rth degrees, in 

 the second auxiliary problem ; so that, by what has been already 

 shown, 



h't = A, - 1, 



h't-2 = ^t-2 + ^t-l + ^t-h ,^33.) 



■}' 



h'^ = h^+ ... + hf- 1, 

 h\ = h^ + h^... + hf—1. 



These last expressions give 



h\ + h>^ + h'^ + ... + h'f = h, + 2h^ + Sh^ + 



>(234.) 

 + thf-t; ^ 



so that the new condition of inequality, (232), may be written 

 as follows, 



m-l>hi + 2h^ + 3hs+ ... + f{hf- l)i . . . (235.) 



and therefore also thus, 



m^ h^ + h^ + h^ + . . . + h( 



+ fi^ + 2 hs + . . . + {t - l){hf - 1). 



I . . . (236.) 



