680 
Proceedings of the Royal Society 
Now, from any one of the remaining six equations, together 
with two equations of the first triad, we can obtain the same result, 
0 = 0. Thus, if the selected equation is b (a -f e) -+- ch - b = 0, then 
from the first and second equations of the triad we have 
(a 2 - e 2 ) + eg - fh - (a - e) = 0 , 
and thence 
(a - e)(6 - ch) + b (eg - fh) - b (a - e) = 0 . 
There is here the linear term b (a - e) - b (a - e), viz., this is 
— U ) { a ' ~ 0 ~ b'(ci - ef\x , 
which is 
= [ — (a - e)ch + b (eg — fh)\x . 
The whole equation divides by the coefficient of x, and gives the 
foregoing equation, 0 = 0. 
Thus the equations reduce themselves to the first triad : writing 
these under the form 
“(a 2 + eg f bd) = *(e 2 + bd + fh) = l(i 2 + hf + eg) = 1 , 
then omitting the last equation (= 1), these are of the form 
U = V = W, where Q, Y, W are homogeneous quadric functions of 
x, y, z-, viz., treating these as co-ordinates they represent two 
quadric cones, having a common vertex, and intersecting in 4 
lines: or we have 4 sets of values of the ratios x : y: z: or for 
x, y, z themselves 8 sets of values; but, as before, these correspond 
in pairs, and the number of distinct solutions is taken to he = 4. 
I return to the equation 0 = 0. This is found to be 
(a - f)x - y, bx cx 
dx [e - f)x — y fx 
gx lix (i - p)x - y 
x = 0 
(p = a + e + i as before); or what is the same thing, the equa¬ 
tion is 
a - 
P 
d 
9 
y 
X 
e - 
P ~ 
h 
y 
X 
f 
p 
y 
l — v — - 
x 
