METHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 343 



of the second degree (194), and h^ equations of the third de- 

 gree (195), to be satisfied by the ratios of m disposable quanti- 

 ties (192). After exhausting, by the general process already 

 sufficiently explained, all the equations of the third degree in all 

 the auxiliary systems, we are conducted to satisfy, if possible, 

 by the ratios of m — /i^ quantities, a system containing 'Aj equa- 

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



'^2 = ^2 + i ^3 (^3 — ')j 



'A, = A, + A3A2 + ^ (^3 + 1) ^3(^3 



-„> } • • w.) 



and after exhausting, next, all the equations of the second degree 

 in all the new auxiliary systems, we are conducted to satisfy, by 

 the ratios of w — Ag — 'Ag quantities, a system of "Aj equations 

 of the first degree, in which, 



-h, = \ + yh^{\-l) (298.) 



We find, therefore, that the number m must satisfy the follow- 

 ing condition of inequality, 



m — h^ — "h^7 "Ai, (299.) 



that is, 



m 7 A3 + '^2 + "^1 (300.) 



On substituting for "Aj its value (298), this last condition be- 

 comes, 



»i7A3+ i'A2CA2+ 1) + Vij; (301.) 



that is, in virtue of the expressions (297), 

 W27Ai+|(A2+l)A2 + i(A2+l)(A3+l)A3 1 ,3Q2. 



+ i(A3+l)^3(^3-l)+i(A3+l)'*3(^'3-l)(/*3-2.)i '^ 



The number m must therefore equal or surpass a certain minor 

 limit, which, in the notation of factorials, may be expressed as 

 follows : 



m < [h, + 1) + i [_K + 1]' + i {K + 1) [^3 + m ,3 . 



+ i[^3+i? + i[A3+i?; J • • ^ '^ 



the symbol [>)]" denoting the continued product, 



[>)]« = ,,(,,_ 1) (,_ 2) ...(«- w + 1). o . . (304.) 



So that when we denote this minor limit of m by the symbol 

 m (Aj, Ag, A3), we obtain, in general, the formula 



m (Ai, A2, A3) = .), + i [rja]^ + hn^ bzf + i [I3]' + B \yi-^\ (305.) 



