where we put 



Then 



As 



we have 



TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 39 



R IL(n—u — v) 



i - 2*.n(2w - 2u - 2v)II(* + u - v) " 



p_ (r+s+%— vy- u l +1 _ (r+s+%— v+r-v,— l) r - u !- 1 



~(s+u-v + l) r - 1 'i +l ~(s+u-v + l + r-u-iy- u i- 1 ' 



2r + s = n; +i-l=-i 



2 > 



_ (w-t'-u-i) r -"/- 1 

 ~ (r+s-vy-"'-^ ' 

 Applying (d) for 



a — n — v — u, 



we get 



From 

 we draw 



b = r— u , 



p_ TX(Zn — 2v — 2u)IL(n — v— u— r + u)H(r + s — v — r + u) 

 2 2r - 2u U(2n-2v-2u-2r+2u)U.(n-v-u)JI(r+s-v)' 



2n—2r = n+s 



r+s = n—r, 



hence the second factor in the numerator, namely, H(n — r—v), when divided by 

 U(r + s — v) gives unity. The third factor above will be U(s + u — v). The first IT in 

 the denominator will be U(n + s — 2u). Thus we get 



r _ U(2n -2v- 2u)U(s +u—v) 

 ~2 n -*-* a IL(n+s—2v)IL(n—v—u,y 



This, by the definition of B x , becomes 



n 1 1 



B 1 2 n ~ sl il(n+s-2v) 

 Hence 



(-l) u +m(2n-2v)Ii(2n-2u) 



ABC = 



2 2 »IMI(w - v)HuTL(n - u)Il(n -s-2u) Jl(n + s-2v)' 

 In the case 2r' = n— 1— s we have 



_ (r' + s+f-t'X""^ 1 _ (2r' + s-u-v + ^y-^- 1 

 ~ (s + u-v + iy- u l +1 ~ (r' + s- v)'"'-"/- 1 

 But 



2r' + s + ± = n-l + i = n--l 



p, _ Q— m-v-I/— '^Tlfe + u — v) 

 II(r'+s — -y) 



Apply (d) by putting 



a — n—u — v, 

 b=r —u . 



Then after a reduction similar to the preceding, 



„,_ II (2-n, - 2u - 2t> )U(n -r' — v)Il(s + u-v) 

 2F^ 2u U(2n-2r - 2v)IL(n -u- v)TL(? + s - v) 



