158 
Proceedings of Royal Society of Edinburgh. [sess. 
On the Ellipse-Glissette Elimination Problem. By E. J. 
Nanson. Communicated by Professor Chrystal. 
(Read January 31, 1898.) 
The problem in question is to determine the equation of the 
curve traced out by any point of an elliptic or hyperbolic disc 
which touches two fixed rectangular axes. Mechanically con- 
structed figures of different forms of the curve have been given by 
Tait,* who also showed that the same glissette can be traced either 
by means of an ellipse or a hyperbola. If p, q are the coordinates 
of the tracing point referred to the axes of the disc as axes of 
coordinates, the glissette is clearly the 0 eliminant of 
(x -p cos 6 + q sin 6) 2 = a 2 cos 2 0 + b 2 sin 2 0 ) 
(y -p sin 6 - q cos 6) 2 = a 2 sin 2 0 + b 2 cos 2 6 J 
and Cayley stated that it would be found to be of order 8 in x, y. 
The actual elimination was first performed by Muir,f who obtained 
in the first instance an equation of order 10. On dividing by an 
extraneous quadratic factor a lengthy equation of order 8 was 
obtained ; and subsequently Lord MILaren j verified the accuracy 
of the terms of highest order in this equation. 
By addition and subtraction we obtain from (1) 
(1, m, n) (cos0, sin#, 1) =0 . . (2) 
(A, B, C, F, G, H) (cos 0, sin 0, 1 ) 2 = 0 . . (3) 
where 
l = 2(px + qy) 
m = 2 (py - qx) 
n — a 2 + b 2 —p 2 — q 2 - x 2 — y 2 
= e 2 - d 2 - r 2 , say 
A = - B = a 2 - b 2 -p 2 + q 2 = A 2 , say 
C — y 2 -x 2 
F = -py - qx 
G —px - qy 
II = 2pq 
* Proc. Roy. Soc. Edin . , xyii. pp. 2-4. f Ibid ., xix. pp. 25-31. 
£ Ibid . , xix. pp. 89-96. 
