METHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 329 



AW = 0, B^*^) = 0, C(^> = 0, . . . TW^ ^ = ; . . (210.) 



h^ + hg + . . . + hf — 1 equations of the second degree, and of 

 the forms 



B^'^) = 0, C^'') = 0, . . . TW =0; (211.) 



0,2 ' 1,2 ^-2,2 ^ ; 



hg + . . . + hf — l equations of the third degree, and of the forms 



CW=o, ...TW =0: (212.) 



0,3 ' t-s,s 



and so on, as far as A^ — 1 equations of the tih degree, namely, 

 TV, = 0,T"o,, = 0,...T^yi) = (213.) 



And third, to satisfy, by the ratio of any one of the m quanti- 

 ties (205.) to any one of the m quantities (204.), this one remain- 

 ing equation of the tth degree, 



T(*') = . (214.) 



For if we can resolve all these three auxiliary problems, we shall 

 thereby have resolved the original problem also. And there is 

 this advantage in thus transforming the question, that whereas 

 there were hf equations of the highest (that is of the ^th) degree, 

 in the problem originally proposed, there are only h^ — 1 equa- 

 tions of that highest degree, in each of the two first auxiliary 

 problems, and only one such equation in the third. If, then, 

 we apply the same process of transformation to each of the two 

 first auxiliary problems, and repeat it sufficiently often, we shall 

 get rid of all the equations of the tih degree, and ultimately of 

 all equations of degrees higher than the first j with the excep- 

 tion of certain equations, which are at various stages of the pro- 

 cess set aside to be separately and singly resolved, without any 

 such combination with others as could introduce an elevation of 

 degree by elimination. And thus, at last, the original problem 

 may doubtless be resolved, provided that the number m, of quan- 

 tities originally disposable, be large enough. 



[13.] But that some such condition respecting the magnitude 

 of that number m is necessary, will easily appear, if we observe 

 that when m is not large enough to satisfy the inequality, 



w > Ai + Ag + ^3 + • • • + ^^? • • • • (215.) 

 then the original h^ + h^ + h^ + . . . + hf equations, being ra- 

 tional and integral and homogeneous with respect to the original 



