679 
of Edinburgh, Session 1871 - 72 . 
which are the required values; there being in all eight solutions, 
but these correspond in pairs of solutions of opposite sign, so that 
the number of independent solutions is = 4. The form of the 
result agrees in a remarkable manner with that obtained by Pro¬ 
fessor Tait on totally different principles (ante, p. 316). 
I annex a further investigation, starting from the assumption 
that the solution is V'M = ad\l 2 + yM 4- z\ viz., writing for 
shortness— 
M2 - ( a', V, o' ) , 
d! 
e', 
/ 
g\ 
h' 
then the solution is 
II 
xa -P 
ya + z, 
xb' 
4 
yb 
xc' 4- yc 
X(P -p 
yd 
xe' 
—i— 
1 
ye 
+ 
+ yf 
xg' -P 
yg i 
xh' 
+ 
yh 
5 
i 4- yi + z 
where observe that only a, e. i contain z; and that the differences 
e-i, i-a, a-e are independent of 2 . We ought to have 
a 2 -P eg -P bd = a 
e 2 + db 4- fli = e 
i 2 4- hf 4- eg = i 
b(a -P e) -)- ch — b 
f (e + i) + dc = / 
g(i + a) + hd = g 
d(a + e) + fg — d , 
h(e + i) + gb = h , 
c (1 -p a) -p bf — c , 
viz., these nine equations should be satisfied by a common set of 
values of x, y, z; or, what is the same thing, the whole system 
should be equivalent to the first triad of equations. To verify this, 
observe that we can from the first triad (by the linear elimination 
of z 2 and z) obtain an equation of the form (x , y) 3 -p x — 0; say 
this is the equation 0=0. In fact, multiplying by e - i, i - a, 
a-e and adding, the three equations give 
(e - i) (i - a) (a-e) 4- fli (e-i) + gc (i-a) + bd (a - e) 
-P a (e - i) + e (i - a) + i (a - e) = 0 , 
where the first line contains terms of the form ( x , y) 3 , the second 
line is linear and 
= \ci(e f - i ■') + e (i' - a') + i (a' - e')]x , 
viz., this is 
— [( e _ i)(i ~ a )( a ~ e ) + fh(e - i ) + gc(i - a) -p bd(a - e)]x . 
The whole equation divides by the coefficient of x, and the result 
is ( x , y) 3 + x - 0. 
