221 
of Edinburgh , Session 1867-68. 
tains the theorems mentioned in the letter of 30th April, and some 
other theorems. It is not necessary that I should particularly 
explain in what manner the present memoir has been, in the course 
of writing it, added to or altered, in consequence of the infor- 
mation which I have thus had of Mr Casey’s researches ; it is enough 
to say that I have freely availed myself of such information, and 
that there is no question as to Mr Casey’s priority in anything 
which there may be in common in his memoir on Bicircular Quar- 
tics and in the present memoir. 
2. Note on the Hodograph. By Professor Tait. 
The object of the present Note is to show, by a few examples 
(of which, however, the last is the only one of any real import- 
ance), how easily the geometrical ideas supplied by Hamilton’s 
beautiful invention of the Hodograph enable us to dispense with 
analytical processes in the establishment of some of the funda- 
mental propositions connected with the motion of a single particle, 
besides many others which are merely curious ; and also how they 
help us to understand the full bearing of some of the analytical 
methods. Some of the simplest of such geometrical investigations 
are given in “Tait and Steele’s Dynamics of a Particle,” and will 
not be reproduced here ; though a few of the results will be 
assumed, — as, for instance, that when the acceleration is directed 
to a fixed point, and varies inversely as the square of the distance 
from it, the hodograph is a circle, and the path a conic section, of 
which the point is a focus. 
1. If the figure represent an ellipse and its auxiliary circle, it is 
known that the circle may be considered as 
the hodograph corresponding to planetary 
motion in the ellipse, but turned through a 
right angle. In fact, if YPZ be a tangent 
to the ellipse at P, SY' is proportional to 
the velocity at P, and perpendicular to it 
in direction. The actual velocity bears to 
SY' the ratio of fx to ha, in the usual notation. 
Hence the tangent at Y' is perpendicular to SP (the direction of 
acceleration), and thus we have an immediate proof that SP is 
