386 Proceedings of Royal Society of Edinburgh. [sess. 
stituting 2 rx, 2 ry, for a and /?, and A -t C - x 2 - y 2 for y, may be 
considered the definite algebraic expression for the glissette of an 
Ellipse or Hyperbola. 
For certain loci of the tracing-point, the eliminant is simplified 
by losing a considerable number of its terms. Comparing the 
original and the abbreviated forms of (1) and (2) of the complete 
equation (p. 383), we observe that 
A = a 2 cos 2 a + 5 2 sin 2 a - r 2 . C = & 2 cos 2 a + a 2 sin 2 a. 
And (1), for A = 0, the locus of the tracing-point is determined by 
u 2 cos 2 a + & 2 sin 2 a = r 2 . 
The locus of the tracing-point in this case is the pedal of the 
generating ellipse from centre as origin of the perpendiculars on 
tangents, and this includes the case of the tracing-point at the 
vertex of the ellipse. 
(2) For A — C = 0, the locus of the tracing-point is determined by 
(a 2 - b 2 )( cos 2 a - sin 2 a) = r 2 . 
When a = 0, the tracing-points are the foci. 
When a > — , r is im- 
possible; and when a<^ the locus of the tracing-point is the 
pedal of a rectangular hyperbola having the foci of the generating 
ellipse as vertices. 
(3) For A + C = 0, we have, (a 2 + Z> 2 )(cos 2 a + sin 2 a) = ?- 2 ; and the 
locus of the tracing-point is a circle concentric with the generating 
ellipse whose radius is + b 2 ). But as J(a? + b 2 ) also measures 
the distance of the centre of the ellipse from the origin, it follows 
that every glissette having its tracing-point on the circle repre- 
sented by A + C = 0, passes through the origin, as is otherwise 
evident from the fact that when y is reduced to — ( x 2 + y 2 ) the abso- 
lute term of the equation disappears. 
The three cases of tracing-point at the vertices, at the foci ? and 
at the extremities of the circumscribing rectangle are noticed in 
the papers of Dr Muir and Mr Hanson on the Glissette Elimination 
Problem ; but it is only by means of the symmetrical expressions 
here found that it is known that these are only particular cases of 
three systems of glissettes, each having a definite locus of the 
