22 DR GUSTAVE PLARR ON THE 



let 2 h 2* k , where h x = n — 2g, 



represent the complexus of points of which the rectangular coordinates are h and k, h 

 being the abscissa and k the ordinate. It is evident that these points will be com- 

 prised within a rectangular triangle OHK, of which the side OH is equal in length to h x ; 

 the side HK will be equal to \\ ; and the hypothenuse OK will be directed along the 



straight line whose equation is 



h-2k = 0. 



We may now conceive two ways in which the enumeration of these points can be 

 made. Either we enumerate them along lines parallel to 0(k) (in columns), or we 

 enumerate them along lines parallel to 0(h) (in bands). 



In the first case each column is given by the operator 



and the sum of the columns will be 



T h k 







(l)=2 \2ffc, 



In the second case each band will begin on the hypothenuse where 



h = 2k, 



2k 



and extend to h = h x , so that 



will represent one band. The sum of these bands, extending from k — to the extreme 

 value of k, namely, i = HK = |^ (or ^^ — 1 if h x be uneven), will be given by 



(2) = T hl k -2 hl h. 



v ' 2* 



Thus the two operators (1) and (2) when applied to W(#, h, k) will give the same 

 results, but (2) is the " interversion " of (1). 

 We have now (replacing h 1 = n— 2g) 



Z n =[t;g ^~ 9) k ?^h]W(g,K,k), 



and we are prepared to introduce the index s in place of h, putting 



h = s + 2k. 

 The limits of s will be 



,s = for h = 2k, 



8 = n — 2g—2k for h = n — 2g. 

 Moreover, we put 



W(g, h, k) = {yy') s 2 cos sty.W(g, k, s) , 



