334 StXTH REPORT — 1836, 



It includes, therefore, in general, the old inequality (215) ; and 

 may be considered as comprising in itself all the conditions re- 

 specting the magnitude of the number m, connected with our 

 present inquiry : or, at least, as capable of furnishing us with 

 all such conditions, if only it be sufficiently dev'eloped. 



[14.] It must, however, be remembered, as a part of such de- 

 velopment, that although, when this condition (232) or (235) or 

 (236) is satisfied, the three auxiliary problems above stated are, 

 in general, theoretically capable of being resolved, and of con- 

 ducting to a system of ratios of the m original quantities (192), 

 which shall satisfy the original system of equations, yet each of 

 the two first auxiliary systems contains, in general, more than 

 two equations of the second or higher degi*ees ; and therefore 

 that, in order to avoid any elevation of degree by elimination 

 (as required by the original problem), the process must in ge- 

 neral be repeated, and each of the two auxiliary systems them- 

 selves must be decomposed, and treated like the system originally 

 proposed. These new decompositions introduce, in general, 

 new conditions of inequality, analogous to the condition lately 

 determined ; but it is clear that the condition connected with 

 the decomposition of the first of the auxiliary systems must be 

 included in the condition connected with the decomposition of 

 the second of those systems, because the latter system contains, 

 in general, in each of the degrees 1, 2, 3, ... ^ — 1, a greater 

 number of equations than the former, while both contain, in the 

 degree t, the same number of equations, namely, h^— \. Con- 

 ceiving, then, the second auxiliary system to be decomposed by 

 a repetition of the process above described into two new auxiliary 

 systems or groups of equations, and into one separate and re- 

 served equation of the rth degree, we are conducted to this new 

 condition of inequality, analogous to (232), 



m-2>h!\ + h\+h\^... + h!\', . . . . (237.) 



A"j, h'\, h"^, . . . h"f denoting, respectively, the numbers of 

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

 second new group of equations ; in such a manner that, by the 

 nature of the process. 



h"t = h', - 1, -) 



A", = h\ + h'^+ . . + h't - 1. 



(238.) 



