155 



1 £____ «^ ^ ^ ?<'+n-l,«+14.2«'_2(' 



rf«'rf/+"-Vz'"'-»' 





" — '^ »!''+«— l,«-f2«'—2(' V 



(U(«)) 



j, (V(S)) 



in which the second members may be calculated by (T!^)). I„ this manner, tlie expressions 

 (R(*>) become, after reductions, 



«(..»') _ [0]-''.[0]-".[0]-"'.2"'.[0]-C'+2+2«'-2(').rHi2 _^ „'_<'"!"' . 



and substituting these expressions in the developments (Q'^^), we get new developments for the 

 constants t/y, uv^/, in which we have only to differentiate for the two variables (a, />) instead 

 of the three {i, z, z'). 

 Again, if we observe that by the nature of the functions F, *, we have when a =: 0, 6 = 0, 



^ ^ dF „ dF d'F 



(W(5)) 



we shall easily deduce relations between the integer numbers t, t', »,, »', which reduce the sum- 

 mation of these developments to the addition of a finite number of terms. For we may prove, 

 either by these equations (W^'), or by the condition (U""') which contains tliem, that the partial 

 differentials 



d' 



t^m' 



r-^6-)' 



da'"dlf" N "•" ' da'"db''" 



Fn. 



rf.*"' 



da J ' 



which enter into the formulae (T'-'')), vanish, unless in the first 



m + m' + 2 > 2(n + n'); 2m + m' + 2 > 3(w + n') ; 



and, in the second, 



w + m' + 2 > 2(n + «') ; 2w -f m' + 3 > 3(n + n'). 



VOL. XV. Z 



