156 



Hence, by (V(^)), the partial differentials which enter into the expressions for m(«''"3, ««(".'>';, 

 vanish unless in the first 



«<<— ''+2; n + n'<:t + l; {X«J) 



and, in the second 



„<t_t'+l; n+n'<:t + l. (YW) 



Thus, in calculating the constants u,,i' , w<,<' , by the fonnulaB (Q(*^), 



we may reject all values of n, n', which are too great to satisfy these relations (X'*)), (YW); we 

 may also, by (VW), reject not only all negative values of the same integers n, n', but all for 

 which the factorial index < + 2 -f- 2»' — 2f is negative in «<"•»'), or <+l+2n' — 2<' in »«("',"'■'. 



and by (R^")), (P'*^) we may reject the value « = in the former, and n' = in the latter. 



Finally, we may remark, that since a factorial vanishes, when its base is less than its index, if 



< + 2 

 both be positive integers, the expression (T(^)) for u("fO vanishes if < be even, and <'> — — ; 



and similarly the expression for to C";"') vanishes if < be odd, and <' > -i- : from which it fol- 



lows, that if the developments (T") of the preceding paragraph, be put by (Y""), (D'^)), (EW), 

 imder the form 



J- (ZW) 



the negative even powers of (#) or of («j) will all disappear. 



Let us verify these general results, respecting the constants «(,(„ w^', by applying them to 

 the particular case t' = t, which as we have before remarked, is the most important in our pre- 

 sent investigations, and which we have already resolved by an entirely different method. In 

 this case, when * = 1 , we find by our present method, 



,,. , — „(1.0) _ Jli_ . w, , = tu{«.i) — ; 



«»•» - "i.i - daMb ' ''^ 1.1 da.db * 



when < = 2, 



d'F d'F d'* 



„,., = „(i.o) +«a.i)=i. >_^^ +1 LJ^I=i 



da^db-^ . ^ ** rfaV^i ^' da^ "^ daJb da.db 



and when f > 2, if we put 



