[815 
815] 
THE BINOMIAL EQUATION X v — 1 = 0. 
73 
SECTION. 
1885), pp. 61—63.] 
tion,” Proc. Lond. 
ent p = 5n + 1, the 
The final result thus is that the coefficients are expressed as functions of the five 
numbers a, ß, f, k, 6, connected by the linear equation a + ß + 6 = ^ (p — 1), and the 
two quadric equations. I remark that formulas equivalent to these were obtained and 
proved by Mr F. S. Carey in his Trinity Fellowship Dissertation, 1884; viz. writing 
n = i(P — 1)> formulae were 
a, b, c, d, e=a — n, ß — n, 7 — n, 8 — n, e — n, 
f, g, h , i, j = ß, e, p, 0-, p, 
k, l, m, n, 0 — 7, p, 8, cr, cr, 
with the three linear relations 
a + ß + 7+ 8 + e = n— 1, 
ß + e + 2p + cr — n, 
7 + 8 + p + 2cr — n, 
and the two quadric relations 
8' 2 + 7 2 + 2cra + (p — a) (8 + 7) — 2p (p + o-) = (8 — 7) (ß — e), 
ß 2 + e 2 + 2pa + (a — p) (ß + e) — 2cr (p + a) = (<y — 8) (ß — e), 
the coefficients being thus expressed in terms of the seven numbers a, ß, 7, 8, e, p, cr 
connected by five equations. The equivalence of the two sets of formulae may be shown 
without difficulty. 
To the Table 2 of the Quintic Equations, given in the paper, may be added the 
following result from Legendre’s Théorie des Nombres, Ed. 3, t. il, p. 213, 
;he periods and of 
he linear relations 
sxamples given in 
ms of the linearly 
or convenience the 
V 
rf 
T 
rf 
rf r; 
1 
641 
1 
+ 1 
- 256 
- 564 + 5238 
- 5120 
-0, 
calculated by him for the isolated case p = 641. 
2/3 — k, -2a-/3-/, 
a, 
ß> 
s in the 20 quadric 
themselves to two 
12 - 2fk = 0, 
: 2 - 2fk = 0. 
C. XII. 
10