376 
Proceedings of Royal Society of Edinburgh. 
15. Taking the four equations 
[SESS. 
Ayh - 2G'xyz + Qx^z = O'l 
Bz^ - 2A'yz^ + Byh = 6 
Cx^z-2B'z^x+Bz^ =0j 
Bxz^ - 2A!xyz + Bxy^ = 0 J 
which are got from the original equations simply by multiplying by 
z and a?, we see that y\ z^, <^x may he dialytically eliminated. 
Doing this we have 
or 
A . (^xh. - 20! xyz 
B B . - 2A'yz‘^ 
P . - 2B' Cxh 
B Bxy"^ - 2A'xyz 
= 0, 
2ABA'yz^ + (ABC + + 2 ARB'^y^ _ (2KPC' + 4AA'B>y^ = 0 . 
From this, by the cyclical substitution, two other equations in the 
same variables are obtained, and therefore only one more is wanted 
for elimination. Now the original equations may he written 
Ay^ - C'xy = C'xy - 1 
Bz^^ - A!yz = A!yz - j. 
- B'zx = B'zx - P^2 J 
and thus by multiplication we have 
or {Ay - Q'x)i^z - h!y){Cx - B'z) = {Cy - (^x){A!z - Ry)(B'aj - P^) 
(ABC + PQE - 2A'B'C').t2/z + 2(CC'A' - QRB')*^^/ + 1:(RB'C' - CAA')a^/ = 0 
Again, from the original equations it is clear that 
2{B'Ry - 3A'B'^ - A!Cx){Ay^ - 2G'xy + = 0 , 
and this, after performing the operations indicated, we find to he 
l^A!B'axyz + i(QRB' - CG'A!)xhj - 2(5RB'C' + CAA') V = 0 . . . . 
But the coefficients of xhj in (i) and (ii) differ only in sign, conse- 
quently by addition we have the equation of the desired form, viz., 
- ^(4RB'C' + 2CA.M)Aj + (ABC + PQR + 16A'B'C')a:2/z = 0 . 
