370 report— 1865. 



form, of which the extreme coefficients are uneven, and of which the determi- 

 nant is a 1 — n, is a root of the equation f a (se, 1 — x)=0, and that this equation 



has no other roots. Again, if % 2 I — , k 2 J =0 is the equation satisfied by the 



squares of the multipliers appertaining to the <[>(«) transformations of order n, 

 the equation \. 2 {n, x)=0 will be satisfied by those roots of the equation 

 f 6 ( x > 1 — fl?)=0 which correspond to quadratic forms of det. — n, but not by 

 the other roots of that equation. For, if <t> a (w) is a root of/ a (a?, 1— .r)=0, 

 <j> s (w) is transformed into vj/"(w) by one of the <f>(n) transformations of order n ; 

 and if 0*(u>) corresponds to a quadratic form of det.— n, the multiplier apper- 

 taining to this transformation is + — =; whereas if 8 ( w ) corresponds to a 



quadratic form of det.— A= <7 ~ - 1 , the multiplier is + [r V-A + io-] -1 (see art. 



T 



131). Forming then the greatest common divisor of the two functions 

 f a i. x > 1— *) and Xz( n > x )' we obtain an equation of which the roots are, exclu- 

 sively, those values of (p s (w) which correspond to quadratic forms of det.— n*. 

 Let 4/^n, x) represent this greatest common divisor, and denoting by jr>\ jV • • 

 the different primes, of which the squares are divisors of n, let us form the 

 expression 





If (A, B, C) symbolize a system of quadratic forms, having their extreme 

 coefficients uneven, and representing the properly primitive classes of det.— n, 

 the roots of the equation "9-Xn, x)=0 are those values of <j> 8 ((*>) which corre- 

 spond to the systems of quadratic equations 



A + 2Bo> + CV=0, C-2Ba,+A w 2 =0. 



Thus the order of the equation is 2h(n) : if x= </>%&) is a root, 1— x=</> 8 { — - ) 



is also a root : the first coefficient is a power of 2, and the last coefficient is 

 unity, because ^(n, x) divides f a (x, l—x), of which the first and last coeffi- 

 cients are respectively a power of 2 and unityf. From the equation 

 ■*i(»> #i) = 0, we ma y deduce two others, <P 3 («, x 2 )=0, %(«, a 3 ) = 0, by the 



1 x 



substitutions x 1 = — , x 2 = — ?—: these equations will have for their roots the 



4h(n) values of f s (w) corresponding to properly primitive forms of det.— n, 

 not included in the subclasses (A, B, C), (C, — B, A). The roots of the 

 equation ¥s( w j ®)=»0 are the reciprocals of the roots of ^(n, a)=0 : its first 

 coefficient is therefore unity and its last a power of 2 ; the equation ^ 3 (n, x) = 

 is a reciprocal equation, and its first and last coefficients are units. 



Each of the three functions ^(n, so), ¥•>(», x), '4' 3 (ji, as), can be decomposed 

 into two factors, of the order h(ii), and containing no irrationality but njn. 



* It is here assumed that if +— j= is not the multiplier appertaining to any of the trans- 



Nn 

 formations of order n by which x is changed into 1 — x, it is also not the multiplier apper- 

 taining to any of the <&{n) transformations of the order n. 



t The limit of / 8 (ar, l-»)-S-ar*<»)+*(»>, when x increases without limit, is 2 4 *W, or 

 _2**(»)+) ( according as n is not, or is, a perfect square. 



