METHOD OF TRANSFORMING AND RliSOLVING BQUATIONS. 341 



condition of inequality, to be satisfied by the number m, in order 

 to the general success of the method (in the case t = 2), 



' m- hz>'h^; (281.) 



that is, 



m>h, + h{K-\- \)K', (282.) 



or, in other words, m must at least be equal to the following 

 minor limit, 



m (A,, ^a) = A, + 1 + i {K + 1) V • • • (283.) 

 For example, making Aj = 1, and Ag = 2, we find that a system 

 containing one homogeneous equation of the first degree, and 

 tivo of the second, can be satisfied, in general, without any ele- 

 vation of degree by elimination, and therefore without its being 

 necessary to resolve any equation higher than the second de- 

 gree, by the ratios of m quantities, provided that this number 

 m is not less than the minor Yimit Jive : a result which may be 

 briefly thus expressed, 



w(l,2) = 5 (284.) 



[16.] Indeed, it might seem, that in the process last described, 

 an advantage would be gained by stopping at that stage, at 

 which, by making i = Ag — 1 in the formulae (278), we should 



have 



^ (A, - 1) ^ 1 1 

 * ^' \ (285.) 



and 



m-i = m-h^+l', (286.) 



that is, when we should have to satisfy, by the ratios of m — Ag 

 + 1 quantities, a system containing only one equation of the 

 second degree, in combination with h^^ + | Ag [h^ — 1) of the 

 first. For, the ordinary process of elimination, performed be- 

 tween the equations of this last system, would not conduct to 

 any equation higher than the second degree ; and hence, without 

 going any furthei*, we might perceive it to be sufficient that the 

 number m should satisfy this condition of inequality, 



m - Ag + 1 7 Ai + I Ag (A^ - 1) + 1. • • • (2870 

 But it is easy to see that this alteration of method introduces no 

 real simplification ; the condition (287) being really coincident 

 with the condition (282) or (283). To illustrate this result, it 

 may be worth observing, that, in general, instead of the ordi- 

 nary mode of satisfying, by ordinary elimination, any system of 

 rational and integral and homogeneous equations, containing n 

 such equations of the first degree, 



