336 



SIXTH REPORT — 1836. 



K248.) 



we find that at the stage when all the equations of the tth degree 

 have been removed from the auxiliary groups of equations, we 

 are led to satisfy, if possible, by the ratios of ?« - h^ auxiliary 

 quantities, a system containing '^, equations of the first degree, 

 '^2 of the second, '^g of the third, and so on as far as 'A^_i of 

 the degree ^ — 1 ^ in which 



'/,, = h, + hfh^ + i(kf+ \)hth^ 

 ■^\{ht-\-2){k,+ l)h,h^+ ... 



+ 2.3.4. .\<-2)^ ^^^+^~^)(^^ + ^~^)--^<(^^~^)^ 



so that, at this stage, we arrive at the following condition of in- 

 equality, 



m - hf> 'Ai + 'Ag + 'A3 + • • + '^^ _ 1, • . . . (249.) 

 'h^,' Agj ' -^hf -I having the meanings (248). In exactly the 

 same way, we find the condition 



m — ht-'hf_-^y "Aj + "Ag + "A3 -I- . . . + "A^ _ 2, • • (250.) 

 in which, 



"'''< - 2 = ''*/- 2 + ^ ^^t _ 1 C^/ _ 1 — 1), 



"-^z _ 3 = '^/ - 3 + '^< - T'^^ - 2 



&c., 



by clearing the auxiliary systems from all equations of the de- 

 gree t — \; and again by clearing all such auxiliary groups 

 from equations of the degree t — 2, we obtain a condition of the 

 form 



(251.) 



U - 3» 



m — hf — ^hf _ J — "A^ - 2 7 '"^1 + • • • + '"^/ 

 in which 



"^^ _ 3 = "^< _ 3 + ? "^/ - 2 C^t - 2- l)j &c- 



(252.) 

 (253.) 



