152 Proceedings of Royal Society of Edinburgh. [sess. 
equations which may be taken to be any three of the six : (1), (2), 
except the sets A, H, H, B, F ; G, F, C. He thence infers, 
first, that the eliminant of any independent set of three of the 
equations (1), (2) is (3), and, second, that A must be a factor of 
the eliminant. He states that by actual development the value 
of the determinant (3) is found to be 2 A 2 , and remarks that it 
would be interesting to show, a priori, that A 2 is a factor. 
4. The ct priori explanation is as follows: — The equation A = 0 
expresses that the line (4) passes through one or other of the 
points in which the conic (5) cuts the line £ = 0. Hence, denoting 
the points in which (5) cuts the lines £=0, rj = 0, £ = 0 by P, P'; 
Q, Q' ; B, B/, the equations B = 0, C = 0 determine (4) as one of 
the four lines QB, QB', Q'B, Q'B', say QB ; the equation A = 0 
then makes QB pass through P or P', and in either case the conic 
breaks up, and so A 2 is a factor of the eliminant. In this 
explanation it has been assumed, as is done by Cayley, that all 
zero values of x or y or z are excluded. 
5. If we regard (3) as the eliminant of (2), a similar a priori 
explanation can be given of the occurrence of A 2 as a factor. The 
equation F = 0 expresses that the two points in which (4) cuts the 
lines ->7 = 0, £ = 0 are conjugate with respect to (5). Hence, 
denoting by L, M, N the points in which (4) cuts the lines £=0, 
7) = 0, £ = 0 , the equations G = 0, H = 0 give L the two conjugates 
M, N, so that (4) touches (5) at L. The equation F = 0 then 
makes M, FT conjugate. But of two conjugate points on a tangent 
one must be a point of contact. Hence M or N must be a point 
of contact distinct from L. In either case the conic breaks up, 
and so A 2 is a factor of the eliminant. 
A similar a priori explanation of the occurrence of A 2 as a 
factor can be given in the case of each of the systems 
B = 0, C = 0, F = 0 ; 
B = 0, C = 0, G = 0 ; 
A = 0, F = 0, G = 0; 
6. Muir, evidently unacquainted with Cayley’s first paper on 
the subject, finds, “ not without considerable trouble,” that the 
value of this dialytic determinant (3) is 2 A 2 , shews that (1) are 
found by expressing that (4) is a factor of (5), and accounts for 
