340 SIXTH REPORT — 1S36. 



in every part of the process. Confining ourselves to the consi- 

 deration of the second auxiliary problem, (which includes the 

 difficulties of the first,) we see that the transformed auxiliary 

 system (266) contains h\ equations of the first degree, and Ag 

 of the second, if we put, for abridgement, 



h'i = hi-l, 1 ^270.) 



A', = A, + ^2 - 1 ; J 

 which new auxiliary equations are to be satisfied, if possible, by 

 the ratios of m — 1 new auxiliary quantities ; so that a repeti- 

 tion of the former process of decomposition and transformation 

 would conduct to a new auxiliary system, containing h'\ equa- 

 tions of the first degree, and h"^ of the second, in which 



h\=zh'^-l, 1 .271.) 



h\ = A', + h'^ - 1, J 

 and which must be satisfied, if possible, by the ratios of m — 2 

 new auxiliary quantities ; and thus we should arrive at this new 

 condition, as necessary to the success of the method : 



m-2>A', + 2(A'2-1); (272.) 



or, more concisely, 



m-2> h\ + h\ (273.) 



And so proceeding, we should find generally, 



m-i>h,^'^ + h^^, (274.) 



the functions A/'\ A^W being determined by the equations 



A w = h„ ^('') = h„ (275.) 



Ag^'+^^-Aa^ = -1, (276.) 



A/i+i)_A,(0 = A^(i + i); (277.) 



which give, by integrations of finite differences, 



Thus, making 



i = K (279-) 



and putting, for abridgement, 



^h, = /i,^^^^ = h, + ^h^{f>^-l), .... (280.) 



we arrive at last at a stage of the process at which we have to 

 satisfy a system of ^h^ equations of the first degree by the ratios 

 of m — A2 quantities ; and now, at length, we deduce this final 



