146 REPORT — 1860. 



represented by / (raj, /(ffr,). /(ra,. 2 ) The periods are the roots of 



certain irreducible equations, each of which is completely resoluble when 

 considered as a congruence for the modulus q ; and the roots u x , u 2 , ... of the 

 congruences are connected with the roots ■sr 1 , sr 2 , . . . of the equations, by a 

 relation precisely similar to that enunciated in art. 44. This relation M. 

 Rummer has established by introducing certain conjugate complex numbers* 

 •*• (raj, ■*■ (ra n ), '¥' (ra',. 2 ), • • • involving the periods only, not themselves divi- 

 sible by q, but each satisfying the n congruences included in the formula 

 ■* (so-,.) (bta-i— «fc) = 0, mod q, 



ft — 1 j ^3 Oj • • • /&• 



From these congruences it is easy to infer that, if/ (ra r , rav, .... ra„ r ) = 

 be any identical relation subsisting for the periods, a similar relation 

 /(«!» w a , ... w») = 0, mod q, will subsist for the numbers u v u 2 ,...u n ; for 

 we find 



* (**»■)/ ( CT '-' w a" . , . ) = ¥ (ra - ,.)/ («u «2 • • • ), mod y, 

 ». e.f(u lt m 2 ,...)s0, mod 9. Another important property of the complex 

 number •*• (Wj) is that it is congruous to zero, mod q, for every one of the sub- 

 stitutions -m x =u^ ■a x — u rv ■w 1 — u r . 1 , • • • except the first : thus the congruences 

 ¥ (u n ) = 0, ■*■ (?«,. 2 ) = are satisfied, . . . but not •*• (u^ — 0, mod q. If, 

 then, t /'(w) be any complex number satisfying the congruence ¥ (ra,.) m /(w) 

 = 0, mod q m , but not the congruence ^ (ra,.) m +'/(w) = 0, mod g ,m+I ,/(w) 

 is said to contain m times precisely the ideal factor of q corresponding to 



* These complex numbers are defined as follows (see the memoir cited at the com- 

 mencement of this article, sect. 3, and that in Crelle, vol. liii. p. 142) :— Let w k be a period 

 satisfying the irreducible equation <p (iir i ) = 0, and let a lt a 2 , ... be the incongruous roots of 

 <p (y) = 0, mod q, b u b 2 , . . . the remaining terms of a complete system of residues, mod q, so 

 that <j> (by), <j> (b 2 ), .... are prime to q. Since -rxr k q = ■Gr /cq , mod q, and ■nr kq = ^sr k , we have, 

 by Lagrange's indeterminate congruence (see art. 10 of this Eeport) 



(«fc"-«i) («*-«,) • • • • {yic-bd O/t-^) . . . . = 0, mod ? , 

 or, since ■ar k —b 1 divides <j> {b r ) etc., 



<P (*i) <P (K) (^ic-Ci) (wj-dj) • • . = 0, mod q\ 



i. e. (uTfc— fli) {^k~ °a) • • • « = 0, m °d 2- ^ Ve ma y now consider the n series of factors 



corresponding to the n values of k [the numbers a x , a 2 , . . . are of course the same for two 

 periods which satisfy the same irreducible equation, but not in general the same for any 

 two periods], and, retaining among these factors only those which are different, we may 

 take for ¥ (isr,) the complex number formed by combining as many of them as possible, in 

 such a manner as to give a product which is not divisible by rj, but which is rendered divi- 

 sible by q by the accession of any one factor not already contained in it. It is evident that 



■*■ (w,) cannot contain all the factors w k — a v ■w k —a 2 , ; let us then denote by tsr k —u k a 



factor which is not contained in M"- (-or J ; we thus obtain the relation 



Mr (to-!) (■vr /c —U k ) = 0, mod q, 



or, changing the primitive root <o into i>) r , 



M' (sr r ) fV ri .---u i )==0, mod g. 



The conjugates of Mr (vrj are all complex numbers formed according to the same law as 

 M' (taTi) itself; and, besides Mr (wi) and its conjugates, no other complex number can be formed 

 according to that law. Also the number u k which corresponds to a given period w k is ab- 

 solutely determined as soon as we have selected the multiplier M*" (•nr 1 ) ; for if two of the 

 factors ■ar k —a 1 , ■ar k . — a 2 , . . . were absent from ^ (nr^ we should have Vr (-nrj (■&■%. — a^ = 0, 

 M r (w,) (-ar k —a 2 ) = 0, mod q; and thence (a x — a.,) M* (tstJ^O, mod q, contrary to the hy- 

 pothesis that a l and a 2 are incongruous, and that Vr (to^) is not divisible by q. The corre- 

 spondence of the numbers u v u 2 u n , with the periods -nr v isr 2 , . . . ■&„., can thus be fixed 



in as many ways as there are numbers conjugate to ¥ (■<*!), %• e. in i-i— different ways. 



