378 Proceedings of Royal Society of Edinhurgh. [sess. 
— that is to say, we have found an axisymmetric determinant loliich 
is the square of one of its own primary coaxial minors. 
To establish this identity directly we have only to subtract from 
the 4th row B' times the 1st row, C' times the 2nd row, and A! 
times the 3rd row ; for by this operation the first three elements of 
the 4th row vanish, and the fourth element becomes 
B A' C' 
A' C B' 
C' B' A . 
17. If, instead of choosing the variables zx^^, xif, xyz^ we had 
taken the similar set x^y, y\ z?x, xyz, the result would have been 
practically the same ; for the set of equations in the latter four 
variables is 
(ABC+PQR)a;22/+ 2APA'^22;+ 2BPC'2;2a;- (2ABB'+4PC'A')a;2/2 = 0 ^ 
2CQA'a;2?/+ (ABC+PQR)y 224 - 2BQB'22a;_ (2BCC'+4QA'B')a;2/2=0 f 
2CBC'a;22/+ 2ARB'?/2z+ (ABC+PQR)22a:- (2CAA'+4RB'C>2/^=0 t 
- (2QRB'+4CC'A')a:22/ - (2RPC'+4 AA'B')y22 - (2PQA'+4BB'C')z2a;+(ABC+PQE,+16A'B'C')a:2/2=0 J , 
and the eliminant differs from that of the preceding paragraph in 
form only, the rows of the one being the columns of the other. 
The explanation of this is to be found in the fact that the 
interchange 
( A, B, C, a:, y, z 
Q, E, P, — 
^ ' X y z 
leaves the original equations unaltered. 
18. It might be thought that the equations connecting a?, y^, z^, 
xyz would lead to a simple symmetrical form of the eliminant. 
This, however, does not seem to be the case, the equations being 
(Q2Rj3 + CBy)x3 + 2ARC'i32/3 + 2BPB'y^3 _ 2A')3y . xyz 
2CQC'cUc3 + (R2py 4 - A2Ca)2/3 + 2BPA'y4:3 - 2B'ya . XlJZ 
2CQB'aa;3 + 2ARA'/32/3 + (P2Qa + B‘^AS)z^ - 2C'aj3 . xyz 
2CQ A'ic3 + 2 ARB'?/3 + 2BPC'z;3 + ( ABC + PQR + 4:MB'C,')xyz 
where a = 4A'2-BC, ;8 = 4B'2-CA, y = 4C'2-AB, and the 
eliminant is consequently of the 18th degree. 
19. Although the various sets of derived equations in the fore- 
going may seem to be obtained from the original set in haphazard 
