476 
a “smith’s prize” paper; solutions. 
[534 
Hence assuming 
0 + a 
0 + a 
= \, we may write 
6+ a 
<j> + a 
A, 
0 + /3_ 
<f> + /3 
6 + 7 _ 
0 + 7 
ï\, 
0 + 8 
$+s~ ’ 
= V(— 1) as usual}, 
viz. this is one of three systems of equations; the other two may be obtained there 
from by writing 7, 8, /3 and 8, /3, 7 successively in place of /3, 7, 8. Hence assuming 
0 + u 
v — -7——, 
0 + it 
the four values of u are a, f3, 7, 8, and the corresponding four values of v are X, - X, 
iX, - iX; and v, u are linearly related to each other; the anharmonic ratio of (a, /3, 7, 8) 
is therefore equal to that of (1, —1, i, —i), viz. we have 
(a — 7) (/3 — 8) _ (1 — i) (— 1 + i) _(1 -i) 2 
(a _$)(£_ ÿ) (1 + i) (— 1 — i) ’ (1 + if ’ 
that is, 
(a - 7) (/3 - 8) + (a - 8) (/3 - 7) = 0, 
or, what is the same thing, 
2 (a/3 + 78) — (a + /3) (7 + 8) = 0, 
= -l, 
viz. we have this relation, or else one of the like relations 
2 (07 + 8/3) — (a + 7) (8 + /3) = 0, 
2 (a8 + /87) — (a + 8) (/3 + 7) = 0, 
that is, the product of the three functions 2 (a/3 + 78) — (a + /3) (7 + 8) 
is = 0. 
But the product in question is (save as to a numerical factor) the cubinvariant J of 
the quartic function; or the equation in question is the required equation J = 0. 
More simply, the linear transformation v = j gives for v the equation v 4 — X 4 = 0 ; 
which is (1, 0, 0, 0, — X 4 $/y, l) 4 = 0; the cubinvariant hereof is =0, and therefore also 
the cubinvariant of the original function (a, b, c, d, e\u, l) 4 . 
Reverting to the equations 
0 + a _ 0 +/3 
(j) + a~ ’ (j> + (3 
-x, 
0 + 7 .. 
,-r 1 = *A, 
9 + 7 
0+8_ . 
t + ■ 
(which, as we have seen, give 2 (a/3 + 78) = (a +/3) (7 + 8)), the same equations give 
0 + a 0 + /3 _ „ 0 + 7 0 +J> _ 0 
(f> + a (/>+/3 ’ <f) + 7 0 + 8