382 
Proceedings of Royal Society of Edinburgh. [sess. 
The equation is now divisible by 1 Qr\x 2 + y 2 ) ; and by putting 
for A and r their values, the equation for focus as tracing-point 
reduces to the sextic, x 2 y 2 {(x 2 + y 2 ) - 4a 2 } + 6 4 (a? 2 + y 2 ) = 0, or 
(g? + ?/ 2 )(icy + $ 4 ) = ia 2 x 2 y 2 . . . . (/) 
This is the same locus as is brought out by Mr Xanson (p. 160 of 
his paper) by a different method of solution. To explain the want 
of an absolute term Mr Xanson points out that the origin is a 
conjugate point, which follows from the fact that in the form 
first given the roots of b\x 2 + y 2 ) are imaginary. 
These results may be verified by finding the equation of the 
glissette for focus as tracing-point directly. If in the original 
equations (1) and (2) we put a 2 -r 2 -b 2 = 0, or C = A, these 
equations reduce to 
a cos 6 = x 2 - A ; /Ssm0 — y 2 -A 
Whence after squaring 
[cos 2 6>] [sin 2 #] 
a 2 
£ 2 
1 1 
[i] 
-(A -a 2 ) 2 1 
- (A -tff =0 
-1 i 
= a 2 (A - $f 2 ) 2 + /S 2 (A - Z 2 ) 2 - a 2 /3 2 = 0. . 
• (/') 
This is identical with the preceding equation when their proper 
values are given to a, /3 , A and r 2 . 
Transforming to axes bisecting the angles at origin, we have for 
the new X and Y coordinates a? 2 + y 2 = X 2 + Y 2 ; 2xy = X . 2 -Y 2 , 
whence 
(X 2 + Y 2 )(X 2 - Y 2 ) 2 - 4a 2 (X 2 - Y 2 ) 2 + 4Zd(X 2 + Y 2 ) = 0, . . (/") 
which is the curve referred to its axis of symmetry. The curve 
consists of two ovals lying on opposite sides of the new axis of X, 
although the mechanical construction only covers one of the ovals. 
By putting Y = 0 the four points of intersection of these ovals with 
the axis of X are given by, X 4 - 4a 2 X 2 + 4& 4 = 0. 
I cannot find that the equation A 0 is reducible below the 8th 
degree for any other tracing-point except the focus. 
