2 
Proceedings of Royal Society of Edinburgh. 
Glissettes of an Ellipse and of a Hyperbola. By 
Professor Tait. (With a Plate.) 
(Read December 16, 1889.) 
Last summer, while engaged with some quaternion investigations 
connected with Dr Plarr’s problem (the locus-houndary of the points 
of contact of an ellipsoid with three rectangular planes) I was led 
to construct the glissettes of an ellipse. I then showed to the 
Society a series of these curious curves, drawn in my laboratory by 
Mr Shand, who had constructed for the purpose a very true elliptic 
disc of sheet brass. I did not, at the time, think it necessary to 
print my paper; but, after the close of the session, I made the 
curious remark that precisely the same curves can be drawn each as 
a glissette of its own special hyperbola. This double mode of sliding 
generation of the same curve seems to possess interest. It is some- 
what puzzling at first, since the ellipse turns completely round, while 
the hyperbola can only oscillate. But a little consideration shows 
the cause of the coincidence. 
Let 0 be the origin, C any position of the centre of the ellipse, 
CA that of the major axis, and P the corresponding position of the 
tracing point. This does not require a figure. 
Then it is easy to see that if </> be the inclination of OC to one of 
the guides, 6 that of CA to the same, we have 
\/a 2 cos 2 0 + 6 2 sin 2 0 = Ja 2 + b 2 coscj> . 
But this gives 
J a 2 cos 2 cf> - b 2 sm 2 cf> = Ja 2 - b 2 cos0 , 
which is the corresponding relation for the hyperbola. In fact the 
one equation is changed into the other by changing the sign of b 2 , 
and interchanging the angles 6 and <£. 
Let the polar coordinates of the tracing point, referred to the 
centre of the ellipse and the major axis, be r , a, we obtain a position 
of P by the broken line OC, CP ; their lengths being Ja 2 + 6 2 , r, 
and their inclinations to the guide <£, 6 + a, respectively. 
If we now turn the guides through an angle a, and use a 
hyperbola whose axes are to those of the ellipse respectively as 
r: Ja 2 - b 2 ’ } and consider the curve traced by a point Q in its 
