TRANSFORMATION OF LAPLACE'S COEFFICIENTS. 



21 



0, 1, 2, &c, ^h (or ^—1 in the case of h uneven), and this for any system of values 

 of g and h. The sum of these results, indicated by 



will then be treated as a function of g and h alone. 



Then, for any value of g we must give to h successively the values 0, 1, &c. n — 2g, 

 and add the results ; we may indicate the sum by 



Finally, we have to sum up 



[K' h ]Wi(9.*) = W&)- 



[2;Vlw<fo) = Z». 



In any multiple sum 2 we may replace an index by another related to it by a given 

 equation. Of course, such an equation must not be dependent on those indices about 

 which the summation 2 have already been effected. 



As our task is to gather together all the terms for which the combination h — 2k takes 



the same value, say s, we can introduce s as a new index ; but as it will be related to 



kby 



h-2k = s, 



we can introduce it only in the place of k and not in that of h at once. 



If we wish to preserve k as an index (a working index) and replace h by s, then we 

 must previously effect an operation which, generally, we will call the " inter-version " of 

 the sums 22. 



This interversion of the order in which two summations are to follow each other 

 requires a corresponding change in the limits of the indices. 



The easiest way to discover what these changes must be consists in giving a geo- 

 metrical meaning to the values of the indices. 



In the present case 



(k) 

































































Sk ,, 



















(h) 





















L 



T 



