86 Proceedings of Royal Society of Edinburgh. [sess. 
The equations of the glissette of the parabola, when expanded, 
are — 
sc cos 0 -jo cos 2 # + ^ sin 0 cos 0 - a = 0 ; ... (a) 
y sin 6 - p sin 2 0 - q sin 6 cos 0 + a = 0 • . . . (b) 
and by addition, 
x cos 0 + y sin 0 -p = 0 ; (c) 
where p = r cos a , and q = r sin a. 
Two new equations are to be formed (1st) by eliminating sin 0 
between (a) and (c), and (2ndly) by squaring (c), viz., 
(py + qx) cos 2 0 - (xy + pq) cos 0 + ay = 0 ; . . (1) 
y 2 sin 2 0 =p 2 + x 2 cos 2 0 - 2 px cos 6 , or 
(x 2 + y 2 ) cos 2 0 - 2 px cos 6 + (p 2 - y 2 ) = 0 ; . . (2) 
By (1st) eliminating cos 2 6 between (1) and (2), (2ndly) eliminating 
the terms independent of 6 between the same equations, and (3rdly) 
eliminating cos 0 between the resulting equations, we find 
{(; V 2 -p 2 ){xy + pq) + 2 apxy)}{ 2px(py + qx) - (x 2 + y 2 )(xy +pq)} = 
{( py + qx)(y 2 -p 2 ) + ay{? 2 + y 2 )} 2 > 
being a function of the eighth degree equated to a function of the 
sixth. 
The elimination of the glissette of the ellipse may be performed in 
the same way, only in this case we have in (a), (b), and (c), instead 
of a and p, quantities containing x 2 and y 2 . Hence for the ellipse 
(1) is of the 3rd degree, (2) is of the 4th degree, the two equations 
immediately derived from these are of the 5th and 6th degrees, and 
the final equation is of the 10th degree. Dr Muir has shown, in 
a paper just read, that the final equation is divisible by a quadratic 
factor, and is thus of the same degree as its limiting form, the 
glissette of the parabola. 
Since the glissette of any curve may be found from the equation 
of the pedal (supposing the latter can be found), the glissette may 
be considered as belonging to a system of derivative curves which 
includes the pedal, the inverse, the reciprocal-polar, and (as shown 
by the writer in a previous paper) the caustic for parallel rays ( Proc . 
Roy. Soc.Edin ., 1 890, p. 280). In this system of derivative curves, the 
curves of each species are defined by a relation between the primitive 
