TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 41 



Now, 



2r=n—s = 2r' + l 



2r+2s = n+s=2r' + l + 2s. 



Hence D and D' are identical, 



D _ D ,__ n(ft— s).Il(n+s-2v) 

 ~ ~ IL{n+s)Il(n-s-2v) ' 



This gives identity in form between ABC'D' and ABCD which represent S(w, v). 

 Thus we have 



where 



S(u ' v)= n(^) R(u)xR(v) ' 



B/ Tf N_ (-l) u n{2n-2u) 



K } ~ 2»II(tt)II(« - u)U(n -s-2u)' 



R(v) = the expression R(u), in which u is changed into v. 



As S(w, t>) is composed in the same manner in u and in v, it follows that 



S(v, u) = S(u, v) , 



and the partition of the values of v in respect to the values of u becomes unnecessary, 

 and we have 



2 u+2 v = Zv. 



M + l 



Returning now to the expression of Z n of § II. , we see that for every value of u the 

 values of v are to be extended from zero to r (or r' , as the case may be), and vice versa. 

 We may then represent Z„ as the product of the three factors 



Bjn-8) , 



(2) [2 r o '\](xr- s - 2u R(u) 



(3) [2 r ( ; r v](x') n - s ^R(v). 



This product, multiplied by 



(yy'Y .2 cos (s^r) 



and summed from s = to s = w, will then give Z„, expressed in function of x, xf, y, y', 

 and v/a 



