TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 37 



We have now in the case 2r = n — s, 



Q 0) becoming Q (,/) in the case 2/ + 1 =n — s. 



On forming the product Q (r) P"(t) we see that Q (,) has in its numerator two factors 



which V"(i) has in its denominator. We have thus in the case of Q 0) , accounting for the 



factor C, 



S(u,v) = ABCT>, 

 where 



" L z 'o l h i i+ 1 (r+s+k-vyi+ 1 (r+8+l-v) i i+ 1 



in the case of Q (,) , containing C, 



S(u,v)=ABC'D', 

 where 



r ,-, (-vyi+nn+i-vy+Hayi+i 



l~ M l'7+i( r ' + s + § - vy+^r' + s + 1 - vy+ l ' 

 If now, for the sake of applying (III.), we put in the case of D, 



a= —v 



S=s 

 we get 



y—r+s+^—v 

 e—r+s+1 —v 



y + e=2r + 2s + %-2v, 

 hence 



£=n — 2r — s , 



which is equal to zero by the definition 



n — s _ 



r= _ , 2r=n — s. 

 2 



We have then 



y — /3 = r + s — n = r + s — 2r— s= —r, 



e- ft = r+.$ + h — n=r+s+%— 2r — s, 

 e-/3=-r + i. 



Thus putting F= 1 in the second member of (III.), we get 



(_ r )»/+i(_ r+ iy»/+i 

 ~(r+s + %- v) v ' +1 (r + s + 1 - v)"l +1 ' 



For D' we have the same values for a, /3, 8, but 



"/=r'+s+%—v 



e' = r' + s + l— V 

 y' + e' = 2r' + 2s + %-2v 

 g = 7l + S + § — 2v 



-2r'-2s-% + 2v 

 — n — 2r — s — 1 , 



