3o8 SIXTH RKPORT— 1836. 



First : to satisfy, by ratios of the m auxiliary quantities 

 «'n • • • «'m' (204) 



an auxiliary system, containing the h^ equations of the first de- 

 gree 



A'i,0 = ^'---Am =0' • • • (206) 

 and the h,^ — \ equations of the second degree 



Second : to satisfy, by ratios of the m other auxiliary quantities 

 «V.--«"m. (205) 



another auxiliary system, containing A, + /<2 — 1 equations of 

 the first degree, 



A'oi = 0,...A^V = 0, ] 



K (261.) 



and ^2 — 1 equations of the second degree. 



Third : to satisfy, by the ratio of any one of the m quantities 

 (205) to any one of the 7n quantities (204), this one remaining 

 equation of the second degree 



B(*-^) = (263.) 



The enunciation of the original problem supposes that 



m7h^ + K\ (264.) 



since otherwise the original equations (193) and (194) would in 

 general conduct to the excluded case, or case of failure, 



«, = 0, ...«^ = (216.) 



In virtue of this condition (264), the first auxiliary problem is 

 indeterminate, because 



»i- 17^1 + ^2-1 (265.) 



But, by whatever sj-^tem of ratios 



-7^,...-^^^ (219.) 



a' a' 



we may succeed in satisfying the first auxiliary system of equa- 

 tions, (206) and (260), we may in general transform the second 



