372 report — 1865. 



<f> H (dw l )=x[ij>' i (w 1 )~], x representing a rational and integral function of which 

 the coefficients contain no irrationalities hut i, sj ' p x , n/jJ», .... The generic 

 equation is therefore Abelian, and of course resoluble by radicals. If the 

 determinant is irregular, having only one index of irregularity, the general 

 expression of the roots of the generic equation is ^(d^d^io,), a nd, as before, 

 it -will be found that H (^i w i)=Xi[ < /^( w i)]> <j>X% w i) = X2[0 8 ( w i)]> tne functions 

 Xi and X2 still containing no irrationalities but i, */pi, Vjj 2 , . . . , and further 

 satisfying the equation Xi[X2(' r )] = X2[xi(' r )]- The generic equation is there- 

 fore in this case also resoluble by radicals ; and the same conclusion is true 

 whatever be the order of irregularity of the determinant — n. If «.=1, 

 mod 4, we should have to consider the generic equation of which the roots are 

 of the form <p s (ui) X ip*(w), and the demonstration of its resolubility is to be 

 obtained in a similar manner. 



We have for brevity confined ourselves to the case of properly primitive 

 forms of an even determinant ; but the values of fj>"(to) corresponding to the 

 improperly primitive classes of a determinant of the form — (4?n-f3), or to 

 the properly primitive classes of even determinants, are also the roots of arith- 

 metical equations possessing analogous properties. A method for the forma- 

 tion of these equations, different from that of M. Kronecker, and not requiring 

 the use of the equation of the multiplier, has been given by M. Hermite 

 (Theorie des Equations Modulaires, p. 44) ; but this method does not supply 

 the ulterior decomposition of the equations into their generic factors. 



138. Application of the Theta Functions to the PeUian Equation. — One more 

 application of the Theta functions to arithmetic (inferior in importance to 

 none that have been mentioned) has also been discovered by M. Kronecker 

 (Monatsberichte, Jan. 22, 1863, p. 44). Let a, b, c, a, r represent real quan- 

 tities, of which a and c are positive and a, b, c satisfy the inequality 

 ac— 6 2 =A>0, and let the positive quantity p be diminished without limit, 

 the limit of the sum 



x = 4- co y = + qo M^t+Tt/) 



2 2 



*= -oo y= -oo la*+2bxy+»f} l+ l 



(in which, however, the value is not to be attributed to x and y simulta- 

 neously) has been found by M. Kronecker to be 



2- 2 



7 IT . 



where 



VA 



log[— 00 + ffw, w)x0(— r-rW, to')], 



(0 



— b+i^A , b+iJA 



w= - — , to = - , 



a a 



6(x, w )=ie u (2Kr, w ), d'(x, <o)= (L6 (*' w) , 



V A being positive, and the logarithms real. 



Again, let o^j — b\=A=ac— b 2 , rtAd being real and a^ positive; and let 

 £ represent an evanescent quantity, we find 



SS [a.v* + 2bxy + cy*] l+ t =2S [a ia * + 2b l xy+c#*] l+ * ' 

 and hence, by the theorem of M, Kronecker, 



