352 



REPORT — 1865. 



2a/.- = r A ~| , , 

 26/, = r B- -J mod T' 



determine the value of 1; because 2« and 2b have no common divisor with the 

 modulus, while the determinant 



2(-TaB + Tb±+<ra)=- [^- 7 2 B 2 + r 2 AC]=2 - = 2y' 



7 7 



is divisible by it ; when h is determined, the equations 



by =2al- - -A, dy' = We - (rB - <r) , 



will supply integral values of c and d ; the matrix ' ^ 

 unit matrix, because 



thus obtained is an 



n n 



it also satisfies the congruence 



a, b 

 c, d 



0,1 

 1,0 



, mod 2, because, from the equa- 



tions (B), taken as congruences for the modulus 2, we find b = c = 1, mod 2, 

 a = d, mod 2 ; but also ad = 0, mod 2, so that a == d = 0, mod 2 ; lastly, the 



equation 



yw + 2/i-_C + rt'<i 



is satisfied by virtue of the first three of the equa- 



y a-\-bu) 



tions (B). Thus to each representation of n there corresponds one, and only 

 one evanescent factor ; conversely to each evanescent factor there corresponds 

 one, and only one representation of n. For, if w annuls the factor \^{u>)— 



<p* ( y ,~ J, the equation (A) and congruence (A') are satisfied by an unit 



matrix \ a ' ■, , in which b>§, but, even if A= —1, by only one such matrix : 



so that the equations (B) determine the values of a and r without ambiguity. 

 The number of factors anmdled by u is therefore equal to the number of 

 representations of n. (ft) Writing ^"(0) for x, we have 



fix, i-.r)=n [V(0)-^y£+^)] =An 1 [f (0)-?V)]> 



where A is the coefficient of the highest power of x in f a (x, 1— x), and IT, 

 extends to every root d>Yw) of f a (x, 1— a')=0, each root having its proper 



f y 8 + 2k\ . , , 

 multiplicity. M. Joubert has proved that, if ^(0) — 9 ( ' — ; — ) vanish when 



0=o>, lim 



is, by the usual rule, 



rd R /yfl + 2A-V 

 ? \ y ) 



is neither infinite nor zero. For this limit 



-1 — 



dti 



_ dO 



.f(0) 





»<M 2 ' 



0=w 



where M is the multiplier appertaining to the transformation w = ^ » 



since (equation 31, art. 125), 



M 2 



