ON THE THEORY OF NUMBERS. 335 



gatlve we find, by a comparison of the formulae (A) and (C), h—h', or. 

 h=3h', according as D^ 1, or ^ 5, mod 8 ; observing only that if D= — 3 

 we have, exceptionally, h^h'. When the determinant is positive, we infer 

 from the formulae (B) and (D), 



,^iogicr+uVD),, 

 iog(T+uvu; ' 



3jogl(TVUVD)^, 

 ""^^ log(T + UA/D; "' 



according as D ss 1 or ^ 5, mod 8. Comparing these expressions with the 

 observations in art. 96 (vi.), we find, if D ^ 1, mod 8, h=h' ; and if D ^ 5, 

 mod 8, A=A', or h=3h', according as the least solution of the equation 

 T* — DU*=4 is uneven or even. 



Dirichlet also deduces from the formulas (A) and (B) the relation which 

 subsists between the numbers of properly primitive classes for any two 

 determinants which are to one another as two square numbers. It is suffi- 

 cient to consider two determinants such as D and DS^, of which the former 

 is not divisible by any square. Ifh and H be the numbers of classes for these 

 two determinants, we have evidently, when the determinants are negative, 



«-s- 



"2 



\n/ n 



\nj n 



the two series in the numerator and denominator not being identical, because 

 in the one n is any number prime to 2DS^ in the other any number prime 

 to 2D. But, by a principle due to Euler, 





p representing any prime, except those dividing 2DS^ or 2D. Hence 



H=.sn(i-(5)1) 



if s denote any prime dividing S but not dividing D. For a positive deter- 

 minant we find 



V V*//iog(T' + UVD/ 



[T', U'] denoting the least solution of the equation P— DS'U''=1 ; i.e. the 

 least solution [Ti, Uj] of P— DU^=1, which satisfies the condition Ui^O, 

 mod S ; so that we may write 



«-|n(.-^)l). 



In a subsequent note (No. 7 in the list) Dirichlet infers from this ex- 

 pression that, given any positive determinant D, we can always deduce 

 from it an infinite number of determinants of the form DS^ having all the 

 same numbers of classes. For if we attribute to S a series of values of the 

 form TT.s", all composed of the same prime numbers s, and having continually 

 increasing numbers for the indices of those primes, it appears from a remark 



to which we have already referred (see art. 96, v.), that the quotient - will 



