TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 



23 



where 



Thus we have 



\y> ^ *) 2 n + s +™llgll(n—g)tt(n -s-2g- 2k)IlkU(s + k) ' 



Z, = [^g t^k T - 29 -™s](yyy.2co S s1rW(g,k,s). 



We will now intervert the treble 2 so as to bring the summation relative to s to the 

 first place on the left. 



(n-2g) 



(in~9) 



Intervening first ~2Jc2s, we have 



•sr.in-9 , ^n-2g-2k v n-2g> v K n - s -2?)i 



Z k z s=z s z * 



oo oo 



Then by a similar figure than the above, 



o# 2 o s = 2 o S2 o 9- 

 We may now write Z„ under the form 



Z n =[T o s](yyJ2co S (s+).z:', 

 where, in putting for abbreviation 



(•') 



we have 



\(n-s) = r, if integer, 



or IO — 1 — s) = r', if integer, 



