84 Proceedings of Royal Society of Edinburgh. [sess. 
reckoned from the centre of the generating curve as origin, and from 
its principal axis as reference-line, and we have, evidently, 
X = X ± r • cos (6 + a) ; Y = /x ± r • sin (6 + a). 
Finally, by substituting for X and /x in (1) and (2), we get the 
simultaneous expressions for the glissette of the two-term oval, viz. : 
X + r • cos (0 + a) = { (a cos 0);r= i + ( b sin ; . (I) 
Y + r • sin (6 + a) = {(a sin 6)f^i + (b cos 0)^}^ ; . (II) 
The equations of the glissette of the ellipse may he immediately 
formed from these by making the exponent n = 2. They are 
X + r • cos (0 + a) = {(a 2 cos 2 0 + 6 2 sin 2 0)}5 ; 
Y + r • sin (6 + a) = { (a 2 sin 2 0 + 6 2 cos 2 0) . 
This method may he further generalised ; because the equation (1) 
is the polar equation of the pedal of the generating curve, and (2) is 
the same equation for a corresponding point in the next quadrant 
of the pedal. Accordingly, whenever the pedal of a curve is known 
or can he found, the glissette of that curve can he obtained from the 
equation of the pedal in the manner above exemplified. 
The equation of the pedal being represented by the generalised 
expression <£ n (R,0,A) = 0, then if the origin of coordinates R,0, he 
taken as the tracing-point, the two equations of the glissette are 
given (in rectangular coordinates, X and /x) by writing in the first 
equation X for R, leaving 6 unchanged, and writing in the second 
equation /x for R, and 
or <£ Jt (X,0, A) = 0 ; <£„ j g x (0 ± ~,A j = 0. 
For any other tracing-point rigidly connected with the generating- 
curve, the first a*-and -y equation of the glissette is derived from the 
polar equation of the pedal by substituting for R, the expression 
x -r cos (6 + a ) , 
and the second ic-and -y equation is formed by substituting for R, 
y - r sin (0 + a) , 
and also changing 6 into (j) ± ^-) 
(III) 
