4 Proceedings of Boyal Society of Edinburgh . [sess. 
This equation suggested, as a useful distance of the tracing point 
from the centre, the quantity 
a 2 - b 2 
2jh 2 + ^ ; 
and accordingly the points 0, A, B, C, D, E, F were taken on the 
corresponding circle. The glissettes of B and D, of course, have 
cusps : — and it is interesting to study the changes of form from one 
to the next of the seven just named. Two groups of figures give the 
glissettes of successive points on each of the axes separately, viz., 
G, 0, K, M on the major axis, and J, F, L, N on the minor. Of 
these K and L have cusps. The figures G, H, J were drawn to 
show how the glissettes of points near the centre approximate to 
the (theoretical) four cusps which belong to the path of the centre 
itself, the finite circular arc described four times over during a com- 
plete rotation of the ellipse. The point P was chosen as close as 
possible to the intersection of the ellipse and the centrode. 
The locus of the instantaneous axis in the guide-plane is of no 
special interest. It is easy to construct it geometrically from its 
polar equation, which may be written generally as 
r( 2 + r) = 4 a 2 b 2 l(a 2 + b 2 ) sin 2 20 , 
or in the present special case 
r(Jba - r) = 4 ct 2 /5 sin 2 20 . 
It is an ovoid figure, symmetrically situated between the guides, 
with its blunter end turned from the origin. 
The equation of the glissettes is found by eliminating 0 between 
the equations 
x = >/a 2 cos 2 0 + 6 2 sin 2 0 + r cos (0 + a) , 
y = V« 2 sin 2 0 + & 2 cos 2 0 + r sin(0 + a) . 
This seems to lead to a relation of the 12th degree in x and y ; but 
it must contain a spurious factor, as Professor Cayley informs me 
the final result ought to be of the 8th degree. And in fact we see 
at once that, if the tracing point be at a very great distance from 
the centre (in comparison with the major axis of the ellipse) the 
glissette will consist practically of four circles, with centres in the 
four quadrants between the guide-lines. 
