ON THE THEORY OF NUMBERS. 



353 



S ■♦'« 



do 



t'V) 



d_ 



Ldo 



* \ y' Jjd = U. 



-dO \ y' )A0 = u> 

 The determination of M is effected as follows : from the equation 

 y b> -f- 21c c -f- du) 



or 



U) 



y a + ho 



_y'c—2Jca+(y'd—2Lb)io 



yff -f y6u> 



it appears (art. 126) that the multiplier corresponding to the compounded 



transformations 



y, 

 -21; y 



and 



a, b 

 e, d 



applied to w, is [<r + i r v/A] -1 ; while 



6-1 



that corresponding to the second of these transformations is simply (—1) 2 i 

 (Table A, ii.) ; therefore — = -( ff + ( > v 'A) 2 , and the limit above written 



<7 -f- /r V A • i . 



— : — — , which is certainly neither infinite nor zero. Hence 



becomes 



tr—ir'/A 



f e (x,l—x) is divisible by x—<p s (io) precisely as often as there are factors 



\p s (0)— W y , ) which vanish when 6=io ; i.e. the multipUcity of the 



root x = (p ■ ( w) in f s (a? , 1 — x) = is precisely equal to the number of solutions 

 of the equation n=o -2 + Ar' 2 , r being positive and uneven. It remains to 

 assign all the values of u>, which, annulling one or more of the factors 



iyd + 2l~\ 



4* s (0) — (j>* \ i — }> give different values to ^"(w). It is evident that values 



" of w, arising from equations associated to properly primitive forms of different 

 determinants, or of the same determinant and different classes, give different 

 values to (p s (w) ; again, of the six subclasses, contained in any one class, the 

 extreme coefficients are uneven in only two ; so that from each class we obtain 

 two and only two values of <p 9 (u>). In the particular case in which the deter- 

 minant is —1, there are but three subclasses, and but one subclass in which 

 the extreme coefficients are uneven ; so that to such a class there corresponds 

 but one value of (f> 3 (<o). Denoting then by A(A) the number of properly pri- 

 mitive classes of determinant — A (we count | instead of 1 for a class of de- 

 terminant— 1), and by (n, A) the number of solutions of the equation 

 « = <r-f Ar 2 , in which r is positive and uneven, we have, for the number of 

 unequal roots of/ 8 (,r, 1 — a;)=0, the expression 



22A(A), 

 and for the whole number of its roots, when each root is reckoned with its 

 proper multiplicity, 



22(«, A)A(A), 

 the summations in each case extending to every value of A, for which the 

 equation w = <7 2 -}-Ar 2 is resoluble with an uneven value of r. 

 We have now obtained the equation* 



2Z(h,A)A(A) = <I>(w)+¥(m), 



* M. Kronecker (Crelle, vol. lvii. p. 250) has exhibited each of the equations I.- VIII. 

 in a similar form. 



1865. 2 b 



