56 
[721 
721. 
FORMULAE INVOLVING THE SEVENTH ROOTS OF UNITY. 
[From the Messenger of Mathematics, vol. vn. (1878), pp. 177—182.] 
Let co be an imaginary cube root of unity, nr + co + 1 = 0, or say co = \ {— 1 + i V(3)}; 
a 3 = — 7 (1 + 3co), j3 3 = — 7 (1 + 3co 2 ), values giving a 3 /3 3 = 343, and the cube roots a, /3 
7 / . 
being such that a/3 = 7; then a + ¡3, = a + -, is a three-valued function (since changing 
the root co we merely interchange a and ; and if r be an imaginary seventh root 
of unity, then 
3 (r + r 6 ) = a + /S — 1, 
3 (r 2 + r 5 ) = wen + &r/3 — 1, 
3 (r 4 + r 3 ) = arct. + n>(3 — 1. 
Any one of these formulae gives the other two; for observe that we have a 3 = — a/3(l +3<w), 
/3 3 = — a{3( 1 + 3<B 2 ), that is, a 2 = — ¡3 (1 + 3«o), /3 2 = — a (1 + 3oo 2 ); hence, starting for instance 
with the first formula, we deduce 
9 (r 2 + r 5 + 2) = a 2 + 2a/3 -4- /3 2 — 2a — 2/3 -h 1, 
= - /3 (1 + 3&)) + 14 - a (1 + 3a) 2 ) - 2a - 2/3 + 1, 
= - a (3 + 3(» 2 ) - /3 (3 + 3a>) + 15, 
= 3<«a + 3w 2 /3 + 15, 
that is, 
3 (r 2 + r 5 ) = <wa + co-f3 — 1; 
and in like manner by squaring each side of this we have the third formula 
3 (r 4 + r 3 ) = co-a + co/3 — 1.