332 
Proceedings of Royal Society of Edinburgh. [sess. 
A less forbidding result is obtained by confining oneself to the 
secondary variables 
x 2 y, 
y 2 z, 
Z 2 W , 
W 2 X 
x 2 w , 
y 2 x, 
z 2 // , 
W 2 Z 
igzw , 
xzw , 
xyw , 
xyz . 
To do so, we perform the operation 
Lx(l) - Bx(4 =) , 
the result being 
BAxy 2 - LD xhj - BAx.w 2 + BG<x 2 w = 0 , 
and three others from it. Next, we perform the operation 
Lz(l)-B Z (4), 
the result being 
L Aifz - Udxyz - B Azw 2 + BG xzw — 0 , 
and three others from it. Lastly, we perform the operations 
«(<*), y(a), z(cl), w( a). 
We thus obtain in all twelve equations; but the resulting deter- 
minant, although now of only the 12th order, is still of the 20th 
degree in the coefficients A, B, C, L, D, etc. 
6. These attempts are neither new as to mode nor satisfactory in 
the result. The same, however, can scarcely be said of the process 
now to be given, where the secondary variables are much more 
complicated in appearance, viz., 
Cx 2 + Az 2 L y 2 + B^ 2 C^; 2 + Az 2 L y 2 + B w 2 
xz * yw ’ xz ’ yw 
or, for shortness’ sake, say 
<f>6, ch , 0 , 
in connection with which it must be carefully observed that the 
result of the cyclical substitution is to change 
into 0, 
and 0 into cf>. 
The equation which has been obtained connecting these variables 
is 
D .00 - 2BG-</> - 2 AE-0 + (4ABK + DEG - D 2 K) = 0 , 
the verification of it being readily effected by substituting for 
