-0 DR GUSTAVE PLARR ON THE 



3. The symbol 



[2>]/(0, 



when the limits a and b are integers, is to represent the sum of the terms (extremes 



included), 



f(a) +f(a + 1) + &c/(a + t) + &c. +f(b), 



f being the representative of a function given in each case. 



When the limits a, b are both fractionary, or one of them only so, then, supposing a 

 the integer value next above a, and b the integer value next below b, the symbol 



K*] /(*) 



will be used for representing the sum or aggregate of the terms 



/(a )+/(a +l) + &c.+/(& ). 



The development of the above inverse square root gives Z n under the form 



Z = r v *V| (-l)m(2n-2g).z n -^ 

 ■ L A> J J 2»nflrII(Ti - g)U(n - 2g) ' 



Substituting for z the expression 



xx' + yy' cos i^ , 



n - 2g |- v « - 2^-[ II(7i — 2g).(xx') n - ig ~ h (yy' COS l/r)* 

 We have also 



by the known formula, in which, for the combination of values of h, k satisfying to 



h-2k = 0, 



the term under the sign 2 has to be halved. 

 We have now for Z„ the treble sum 2, 



z n =[^;gT- 2 %^ o h k]Wg,h,k), 



in which, after reduction, we get 



w . , ( - l)m(2n - 2g)(xx')."-*«-\yy'y 2 co s (h - 2k )^ 



u. . p; 2«+ A n((/)n('M-^)n(u-2(/-/on(A;)n(/6-A;) ' 



In any multiple sum 2 the summations are to be effected from right to left — that is 

 to say, in this present case of a treble 2, we must give to k successively the values 



we get 



r n-2g -i T" 



= L 2 ° h * ~ U(h)U(n-2g-h) 



