TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 27 



First, T g J, u v = 2 u v 2 V ; 



Secondly, 2> Jf^v = t^v 2> 



Thus we get 



z (4) =r 2> Tv ?q ?i -f n ~°~ v ~°~ 28w '" 



r v^ 2^ 2.1 n 



L +2^2> 2» 23 J 



We may write this under the form 



2 ^ 2,, +1 v ]» « S(v,u), 

 where 



S(u,i;)=[2>2''nW'" ) 



L "« J i 



and S(v, u) is derived from S(w, v) by the permutation of u, v. 



These double sums represented by S(w, v), S(v, u) depend on the indices alone 

 being freed from the variables x, x' . They are therefore simply numerical, depending of 

 course on n, s, u, v ; r being the abbreviation of \{n - s) when integer, and being 

 changed into r' when \{n — 1 — s) is integer. 



We introduce two new indices in the place of g, I, by 



g = v-i 



l = w+j. 

 As we have 



we do not change the sum in taking the elements relative to i in inverse order. Thus 

 we have now 



where by the expression of W //r at the end of § I. we have 



w ,„ _ (-iy+ i U(2n-2v + 2i)n(u-v + i+j)2 2v '- 2i 

 ~2 n + s + 2u +VIl{v-i)Il(n-v + i)Il(n-s-2u-2j) X 



1 



IT(s + u - v + i +j)Il(u -v + i)Il(j)IL(i)TI(u —v+j)' 



We will now make use of factorials in order to extract from W /f a factor A inde- 

 pendent of i, j. 



VOL. XXXVI. PART I. (NO. 2). F 



