1898 - 99 .] Lord McLaren on Glissette Elimination Problem. 383 
The determinant for a tracing-point on the minor axis is found 
similarly from 
x — rsin0= J(a 2 Q,o& 2 0 + & 2 sin 2 #) 
y - r cos 0= J(b 2 CGS 2 0 + a 2 sin 2 0) j 
II. Expression of the General Equation of the Glissette as a 
Symmetrical Determinant. 
(A); 
The simultaneous equations of the glissette for any tracing-point 
rigidly connected with the generating ellipse are 
x — r cos (6 — a) = J(a 2 cos 2 0 + & 2 sin 2 #) ) 
y - r sin (0 - a) = J(b 2 o,os 2 0 + a 2 sin 2 #) j 
where r is the distance of the tracing-point from the qentre of the 
ellipse and a is the inclination of the line, r, to the major axis. 
A symmetrical bordered determinant can be got from the 
equations in this form, but its expression is much simplified by 
putting cf) = 0 - a; cf> + a = 6 ; or 
x-rcos<f> = J{a 2 cos 2 (cf> + a) + 5 2 sin 2 (<£ + a)} ) 
y - r sin <£ = J [ b 2 cos 2 (<£ + a) + a 2 sin 2 (<£ + a) } J 
By squaring each side and expanding (<£ + a) we get, 
j " a C ° S . a X cos 2 cf> + 2 (b 2 - a 2 ) cos a sin a cos <£ sin <£ + i" ^ C0S a X sin 2 </> 
( + 6 2 sin 2 a - r 2 J ( + a 2 sin 2 a J 
+ 2racos <£ — cc 2 = 0 . . (1) 
i ^ CO f a 1 cos 2 (f> + 2(a 2 — b 2 ) cos a sin a cos sin <j> + < a C0S a 1 sin 2 <£ 
( + a 2 sin 2 a J ( + 6 2 sin 2 a - r 2 ) 
+ 2ry sin <£— y 2 = 0 . . (2) 
By addition, 
2rx cos <£ + 2ry sin </> + a 2 + b 2 - r 2 — x 2 — y 2 — 0 . . (3) 
These expressions, with abridged coefficients, are equivalent to, 
A cos 2 <£ + 2B cos sin </> + C sin 2 <£ + a cos <£— x 2 = 0 . (1) 
C cos 2 <£ - 2B cos sin <£ + A sin 2 <£ + /3 sin - y 2 = 0 . (2) 
By addition, acos<£ + /?sin <£ + y = 0 . . . (3) 
where y = A + C - x 2 - y 2 . 
We have also, cos 2 <t> + sin 2 ^> — 1 = 0 . . . -(d) 
