330f REPORT— 1861. 



If be the rath term of the series S , in which we 



suppose that the terms are so arranged that no term surpasses any that precedes 

 it, it can be shown that lim^=:^!^^^ ''. For if SA^+^a, SAij + r;^ represent 



generally any one of the 2Ai//(2A) systems of values that can be attributed 

 to X and y consistently with the condition that (a, b, c) assumes a value 

 prime to 2A, the number of terms up to A-„ inclusive {i. e. the number n) is 

 evidently equal to the number of points having coordinates of any one of the 

 forms [2A^+^g, 2Aij + ?yJ that lie within the ellipse aoir + 2bxy-{-cy-—hn, 

 together with one, or all, or some of the similar points lying on the contour 



of the ellipse, according as is the first or the last, or neither the first nor 



the last of the terms equal to it in the series. The area of the ellipse is 



Trie 



-y-i ; whence, if n be very great, the number of the points we have defined 



is approximately L^ " the error being of the same order as V^n ; 



4A-V A 



i.e. lim I?.=J!^K2A) 



Z;„ 2Av'A* 



Hence by Dirichlet's first Lemma (art. 99) lim pS= , . h. Again, by 



1 vI'(2A) 



the same Lemma, the expression pS -jqr lias for its limit, when p 



diminishes without limit. And, lastly, the limit of the series S (— j-fij- is 

 the series 2 ( — ) — , in which the terms are taken in their natural order. To 



establish this, we observe that the symbol f — ) is a periodic function of n, 



and that the sum of the terms of which one of its periods is composed is zero. 

 Using the notation of art. 98, and attributing the value + 1 or —1 to the 

 symbol 2 according as P^ 1 or ^3, mod 4, and to the symbol e according 

 as D=PS^ or =2PS^, we have, by Jacobi's law of reciprocity, 



/rv\ "— 1 «"— I / \ 



Hence ( — )~( ~ )> if « ^ w'j niod 2*PR* ; or ( — I is a periodic function of w. 



Again, if a and b denote the general terms of a system of residues prime to 

 2* andj9 respectively, we find 



sa 2 g 8 /pWss 2 ^ 8 xn.2(^jxn(r-i), 



where in the left-hand member the summation extends to every value of « 

 prime to 2*PQ and less than it, while in the right-hand member the signs of 

 summation refer to a and b, and the signs of multiplication top and r respec- 



* The index k is not the same as in art. 98 j it is 1 when 5=1, e=l ; 2 when S=- —1, 

 6=1 ; and 3 when e = — 1. 



