ON THE THEORY OF NUMBERS. 327 



only the first is convergent ; while the two series 



11 11 I 



3 2^5^7 6 

 are both convergent, but have two very different sums*. 



These observations will show the importance of the following prpoosltionf. 

 " If c„ be a periodic function of n, satisfying the equations 



C1 + C3 + C3 + .. +CA=0, 



the series S — in which the terms are taken in their natural order, is con- 



w=l 

 vergent for all values of * superior to zero, and its sum is a continuous func- 

 tion of s." 

 For if we add together the k consecutive terms 



we obtain a fraction of which the denominator is of the order ks in respect 

 of »i, while the numerator is only of the order (k — l)s — 1, because the co- 

 efficient of m(*~'^* is zero. We may therefore replace the given series by a 



series of the form S , in which 0(»i) is a function of the order 1 +« 



in respect of m. This series is always convergent for positive values of « ; its 

 convergence is irrespective of ihe arrangement of its terms, and its sum is a 

 continuous function of *, because <l>(m) is a continuous function of *. The 

 given series is therefore also convergent, and its sum is a continuous function 

 of 5. 



100. The second section of the memoir refers to the symbols of recipro- 

 city of Jacobi and Legendre (arts. 15, 16, and 17 of this Report). 



The third and fourth sections contain the principal theorems relating to the 

 generic characters of quadratic forms, and to the representation of numbers. 

 There is only one of these theorems to which we need direct our attention 

 here, as the others have already come before us in the preceding articles. 



Let (a, b, c) be a primitive form of the positive determinant D ; (a, b, c) 

 (Xg, y„)-=M a positive number represented by (a, b,c); m the greatest com- 

 mon divisor of a, 26, c ; [T, U] the least positive solution of T*— DU^=m2 ; 



so that if ir„=- [T„3:o— U„ (5xo+cy„)], y„=_[T„y(,+U„(aa:o+Jyo)], 



the two formulae \_Xn, yn\ and [ —Xn-, — y«] will together express every repre- 

 sentation of M, which belongs to the same set as [^0,^0]. Similarly, let 

 [■^''»i>y»]> [ — x'n, — ^'„] denote a complete set of representations of the posi- 

 tive number M' by (a, h, c). 



If we trace the hyperbola represented by the equation aa:'+26a2/+cy'=l 



referred to rectangular axes, the diameters included in the formula y= ~ x, 



in which h is to receive all values from — 00 to +00, will form a pencil of 



* These illustrations are taken from the Memoir on the Arithmetical Progression in the 

 Berlin Memoirs for 1837, pp. 48 and 49. 



t The demonstration in the text is a little simpler than that given by Dirichlet, who uses 

 the function r to express the sum of the series. 



