1889 - 90 .] Professor Tait on Glissettes of an Ellipse. 
3 
plane, whose central polar coordinates are Ja 2 + 6 2 , - a ; the position 
of the point Q is given by the broken line OC',C'Q. Of these OC' 
is equal and parallel to CP, while C'Q is equal and parallel to OC. 
Thus the points Q and P coincide. 
In fact the motion of either is the resultant of two circular 
motions, one of which is complete (viz., 0 , which has all values from 
0 to 2tt), the other reciprocating (viz., <£, which varies "between 
sin - \bj ' J a 2 + b 2 ) and sin -1 (a/ Ja 2 +b 2 )). But, in the case of the 
ellipse, the centre has the reciprocating motion; while, in the 
hyperbola, it describes the complete circular path. 
Mr Shand has constructed a hyperbolic disc, comprising a con- 
siderable portion of each of the branches of the curve, and it gives 
very fair glissettes. It is very curious to watch the proper point of 
the hyperbola gliding over the curve already traced by the ellipse. 
But this apparatus is not so easily managed as is the elliptic disc, so 
that the figures in the plate were drawn by means of the latter, and 
reproduced on a diminished scale by photolithography. 
To exhibit, by a few forms, as completely as possible the general 
nature of these glissettes, I selected a series of tracing points equi- 
distant from the centre of the ellipse, and situated within and on 
the boundaries of the various regions, to each of which belongs 
a special form. For this purpose I traced the curve formed by 
successive positions of the instantaneous centre of rotation on the 
disc. The disc, with this curve on it, is represented in the upper 
central figure. The equation of the curve is 
b 2 x 2 - a 2 y 2 _ Ja 2 + b 2 Jx 2 + y 2 
b 2 x 2 -f a 2 y 2 a 2 - b 2 
It is easily traced as follows. Draw the ellipse whose semiaxes 
parallel to x and y respectively are 
a 2 -b 2 j a a 2 - b 2 . , 
Ja? + b 2 an b ^ a 2 + ^ 2 ^ 
diminish every radius vector in proportion to the cosine of double 
the angle vector ; and then diminish the ordinates in the ratio b : a, 
so that the ellipse itself becomes a circle. 
In the disc from which the glissettes were drawn, a (rather more 
than a foot in length) was made double of b. 
