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 sajne result, 
0 = 0. Thus, if the selected equation is b (a + 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)(b - ch) + b (eg - fh) - b (a - e) = 0 . 
There is here the linear term b (a - e) - b (a - e), viz., this is 
= \b(af - e) - b\a - e)~\x , 
which is 
— [ - (a - e)ch + b(cg — 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 -t- eg -h bd) = -(e 2 + bd + fh) = l(i 2 + hf -f eg) = 1 
then omitting the last equation ( = 1), these are of the form 
U = V = W, where [J, 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, ^ themselves 8 sets of values ; but, as before, these correspond 
in pairs, and the number of distinct solutions is taken to be = 4. 
I return to the equation 0 = 0. This is found to be 
(■ a - p)x - y, bx cx 
dx (e - p)x — y fx 
gx hx (i - p)x - y 
- x = 0 
(p = a + e + i as before) ; or what is the same thing, the equa- 
tion is 
a — p - 
= 0 . 
e - p - 
9 
h 
> * - P 
