774 : REPORT—1863, 
These congruential determinations possess great interest, not only because — 
direct ‘methods. of solution present themselves very rarely in the theory of 
numbers, but also on account of the singular connexion which they establish 
between certain binomial coefficients and certain quadratic decompositions of 
primes. Nor is it less remarkable that the properties of the resolvent func- 
tion of Lagrange form the intermediate links in this connexion; although it 
is proper to observe that Gauss has exhibited his demonstration of the theorem 
relating to the equation p=a*+ 7’ in a form in which its connexion with the 
theory of the division of the circle is disguised. 
Results of a more general character haye been obtained by Jacobi and 
Cauchy. Cauchy has treated of the subject with great fulness of detail in his 
Memoir on the Theory of Numbers, in the 17th volume of the Memoirs of the 
Academy of Sciences (pp. 249-768); while Jacobi has barely indicated his 
method in his note on the division of the circle (Crelle, vol. xxx. p. 166); 
nevertheless, as in some respects it seems preferable to that employed by 
Cauchy, we shall endeavour to adhere to it in what follows. 
‘Retaining the other notations which we have employed in this article, let 
ae =wv(m, n, 6) or (m,n), When there is no occasion to consider 
@ explicitly ; we observe that ~ (m, n)=wW (n, m); W (0, n)=u (m, 0)= 
W (0, 0)=—1; also W (m', n')=y (m, n), if m' =m, mod p—1, n' =n, 
mod p—1; W (m,n)=(—1)"** p=(—1)""! p, ifm+n=0, mod p—1, but m 
and n are not =0;mod p—1. Letm,,m',,....m,‘” be any set of +1 num- 
bers, each of which satisfies the conditions 0 <m, <p—1; let m,4+m’',+... 
+m,=n, (p—1)+5,, where 0 =s,<p—1; and put F(0-™) F(@-™)... 
F (9-™») = (6) F (6-"1.) ‘Writing, for brevity, 
K=n, ph S=m4tm,, pn" =m,+m,'+m,",...modp—1, 
and determining j1,, 4,', 4," -..80 as to satisfy the conditions 0 < p, <p—1,we 
find x (8) = (sy, m,") W(p', ,")....W(u,-?, m,). In this expression if 
fO +m") > p—1, we write for W (u,%, m,“*”) its equivalent p+ 
W(p—1—p,©, p—1—m,”); and if p+m,“t?=p—1, we write for 
W (py? 74°F) its equivalent (—1)ttm p. It is evident that the condi- 
tion ph +m°+) > p—1 will be satisfied n, times precisely ; so that x (6) 
assumes the form p71 aes ©, (6) and ¥, (8) denoting products of factors of 
the form (h, h’), in each of which h+h'<p—1. It will now be found 
©, Gy) = IIs ; 4 : 
that: 2004 == (—1)e-m_ ee _ mod yp. For (1), fa (+1) 
{eG AT oe 
<p—1, we have p (u,, m,“*, y) = — Oy, *” mod p; (2) if n,@ 
’ By 3 1 2) Bie? [ Tm, 6” , Pi 
+m,“t) > p—1, we have $$ 
‘i v(p—1—p,°, p—1—m,“*, y) 
_ H(p—1—p,©).0(p—1—m,) _ np,“ Mee 
Op —2— pam) = Bay tm or? OOP 
—1)9 Pes 
since, by Sir J. Wilson’s theorem, [I (p—1—j) = = mod pif <p—1; 
(3) if p, +m,°* =p—1, we have 
