:rHOD OF TRANSFORMING AND KKSOLVING KaUATIONS. 345 



m (A„ h^, h^, h^) = \ + /*! 



+ — ^2 (12 + to h, + 9 V + 2 V + S V) 

 24 



+ lh^hs{\ + h, + h-) +lh^h,^+ Ih, 



+ ± h^ (20 + 22 ^4 + 25 A/ + 9 V 

 48 



+ 8 ^4^ + 5 ^4" + 3 ^4^ 



+ 1. h^^ (18 + 10 A4 + 15 A42 + 14 A43 + 9 A44) 

 48 



+ 1^V(1 +3A4 + 3A4^) + -^A34 



+ _L (432 A4 + 364 A42 4- 108 ^4^ + I69 h^^ 



)> . (315.) 



J 



+ 24 A4* + 34 A4« + 12 h^' + 9 hi). 

 As examples, whichever formula we employ, we find 



(316.) 



(317.) 



(318.) 



m(l, 0, 1, 1) = 7; 



m(l, 1, 1, 1)= 11; .... 

 m(l, 1, 1, 2) = 47; .... 

 m {5, 4, 3, 3) = 922 (319.) 



[19.] In general (by the nature of the process explained in 

 the foregoing articles) the minor limit (256) of the number m, 

 which we have denoted by the symbol 



7n(Ai, Ag, . . . A^), 

 is a function such that 



m (Ai, Ag, . . . A^) = 1 + m (A'„ A'^, . . . A'^), . . . (320.) 

 A'l, . . /I'f being determined by the formulae (233). This equa- 

 tion in finite differences (320) maybe regarded as containing the 

 most essential element of the whole foregoing discussion ; and 

 from it the formulae already found for the cases t = 2, t = 3, 

 f = 4, might have been deduced in other ways. From it also 

 we may perceive, that whenever the original system contains 

 only one equation of the highest or rth degree, in such a man- 

 ner that 



A,= l, (321.) 



then, whatever t may be, we have the formula 



