1898 - 99 .] Lord M‘Laren on Glissette Elimination Problem. 387 
tracing-point, and a definite equation which is less complex than 
the general equation. The case of the tracing-point on one of the 
axes represents a fourth system. Except in the unique case of the 
focus, where A - C — 0 and B = 0, the reduced eliminant is not 
divisible by a factor, but remains an equation of the 8th degree. 
III. Formation of a Symmetrical Determinant from the 
Original Untransformed Equations. 
As already stated, a bordered symmetrical determinant may also 
be obtained from the original simultaneous equations without 
changing 0 into <f> + a. (A, p. 383.) The partial eliminants are 
formed in the same way as before, and I shall, therefore, only 
write down the resultant. 
The original equations being expanded in powers of 9 , we put 
A, B, C, cq, p v for the coefficients of cos 2 0, cos 9 sin 0 , sin 2 (9, cos 0, 
and sin 6 in the first equation ; and C, — B, A, a 2 , /? 2 , for the co- 
efficients of like powers in the second equation : (a v /3 2 ) stands for 
(cq/^ — a 2 /? x ) according to the usual determinant notation, and the 
resultant is, 
(&,«,) + 2By , (C-A)y , A(C -y*)-p 2 (A-x*)\ 
+ !>(«! + a 2 ), j a l +a 2 
(C-A)y , (^ 1 ,a 2 )-2By , a 2 (C - iC 2 ) - aj(A -y 2 )\ 
-£(&+&),/ ft+/ - 2 
(/?l, a 2 ) , y 
y 
This expression only differs from A x in the values of the co- 
efficients, and in the circumstance that the first powers of sin 0 
and cos 6 occur in each of the two original equations, and come 
out in duplicate in the resultant. 
As this resultant can be formed from one of the original quad- 
ratic equations and the linear equation of addition, and as these 
two equations contain nine distinct coefficients, it follows that in 
the elimination of sines and cosines between any two equations 
of the 1st and 2nd degrees respectively the resultant is a bordered 
symmetrical determinant of the 4th order. 
ft(C- 2 / 2 )-ft(A-«r 2 ) | a 2 (C —x 2 ) — a, (A — |/ 2 ) 1 
+ L(a x + a 2 ), j - B(p 1 + fi 2 ), j 
«! + a 2 > /?i + /?2 j 
