TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 35 



We have now 



S(^)=A.B. [2 r ; u J 2 ^]QO')P'(ii). 



We may intervert again the sums 2, because the limits of the indices are independent 

 from one another. 



In the denominator of QP' we remark that the product 



(s+u - v + iyi+i x(s+u - v+j + 1)' 7+1 

 can be written under the form 



+ u - v + iyi +i x (s + u - v + i + iy/+ x . 



By these changes we get 



S(u,v) = A.B. [%a r ; u j] F'(i)Q'(i,j), 

 where 



P , Y .v _ (-vyi+K(n+^-vyi+ i .(syi+ 1 



r W iif + i ^ n + i_ u _ v yi+i ,( s+u _ v + iy/+ 1 



/ n—s V' /+1 / n-l-s , V /+1 



V'+\s + u-v + i + iyi +1 

 To the summation of 



we can now apply (I.), but we must treat separately the two cases 



(1) n-s=2r, (2) n - 1 — s = 2r' , 



r and r' being integers. 



In the first case we have to put 



a=-r + n, /3= -r + ^ + u 

 /3 = a + | y=s+u- v + i + 1 , 



and we get 



o ry-vi cm i) - (r+B+i-v+jy-"+ i 



In the second case we have to put 



, / n - 1 - s . \ 

 a=y g h u J = -r' + u, 



then /3' = a' - \, namely, 



VOL. XXXVI. PART I. (NO. 2> O 



/3' = l-^+W=-r'-l+ W 



