ON THE THEORY OF NUMBERS. 373 



lim [~22 1 — 22 1 I 



L [aa?+2bwy+ctff+e [a 1 o?+2b 1 a ! y+c i y*f+<!j 



e i- v«_i e 



=lim 22-——- : — ^tt-„-22- 



[oa? a + 26«y + cy 2 ] ' + f [«^ 2 + 2b x xy + c^f] ' 



+0 



_ 2tt lo „ flVfffl'(0.^) e '(0, q»i') ffi v 



3</a ° fl,v«, e'(0, «)e'(0, «') ^ ; 



This result M. Kroneeker has applied, in the following manner, to the solution 

 of the Pellian equation. 



Let P and Q, be positive integers not divisible by any square, of which P 

 is >1, and let m and n represent positive integers prime to 2P and 2Q 

 respectively, the limits of the sums 



2 (£)_i_and 2 (— ^)_ 

 m=l W™ 1+ * n= l\ n J n l+ § 



are known from the researches of Dirichlet (Crelle, vol. six. p. 360 and 364 ; 

 or art. 101 of this Report), and are respectively 



*(P)lo g[T + ITVP] and g*(-Q) 



2VP 2VQ ' 



7((P) and /<(— Q) denoting the number of properly primitive classes of the 

 determinants P and — Q, and T, U being the least positive numbers which 

 satisfy the equation T 2 — PU 2 =1. Multiplying the two results together, and 

 designating PQ by D, we find 



4^ 7 < P )- 7 <- Q > 1 °8< T + IJ VI > )=lim22^y^_ 1 _. (Hi) 

 4VD V»/\ * /(mil) 1+ ? 



If P and Q, are relatively prime, and congruous to one another, mod 4, so that 

 D is not divisible by any square, and is = 1, mod 4, the scries 



'LMb)^] 2 (s) ss ( ^ +2 4 +w/ )i +? • • • (A) 



(iii which R=P, or E=Q, according as P=Q = 1, mod 4, or P=Q=3, 



mod 4 ; / or (a, b, c) denotes any one of a set of representative forms of 



det.— D ; (-LA is the particular character of /with respect to'R, and the first 



sign of summation extends to every form of the representative system) is 

 identical with the series 



22(*)(=Q)-1— ( B) 



To verify this we observe (1) that, because D==l, mod 4, the numbers of sets 

 of representations (art. 87) of N and 2N" by forms of det.— D are equal, N de- 

 noting any number whatever ; (2) that, because D is not divisible by any 

 square, the number of sets of representations of N by the forms of det.— D is 



2 ( — — \, N denoting any uneven number, and d any divisor of N which is 



prime to D; (3) that the generic character of a form /of det.— D may be 

 ascertained from any number whatever N which is represented by f ; in fact 

 if p is a prime divisor of D, and if N"=Fp 2 ", N' being prime top, we have 



(-)=(— ) ; if N=N'^"+', D=Dp, W and D' being prime to p, 



