446] 
OF A NODAL BICIRCULAR QUARTIC. 
187 
which show that the cuspidal curve is the inverse of a conic (viz., of an ellipse, if, 
as in the figure, m is positive). The result in the very same form would be obtained 
by considering the curve as the locus of the vertex K of the variable triangle. 
If we imagine a plane rigidly connected with the link AA', and carried along 
with it, then (6, c) are the coordinates of the point G in this moveable plane ; and 
if, as above, (a, fi) are the coordinates of the node, then (6, c) and also (a, fi), are 
given functions of (u 1} u 2 ). We have thus (b, c) functions of (a, fi), and reciprocally 
(a, fi) functions of (b, c) ; that is, we have a correspondence between the points of the 
fixed plane and those of the variable plane. It is worth while to investigate the nature 
of this correspondence, although the result does not appear to be one of any elegance. 
Writing 
A _(m+V)a 
m+ l)a a. 
m a 2 + fi 2 ’ 
m 
B _ (m + 1) a /3 
m a 2 + fir ’ 
we may, in place of (a, fi), consider the point in the fixed plane as given by means 
of the inverse coordinates (.4, B). And then, if p = u 1 + u 2 , q = l — u x u 2 , we have 
m + 1 — q’ m + 1 — q 
whence 
Hence 
which determine (b, c) in terms of (p, q); that is, of (A, B) or of (a, fi). 
In reference to some other constructions given in Mr Roberts’ paper, it may be 
remarked that if we have a moveable plane Hj always coincident with a fixed plane 
n, and if a condition of the motion is that a circle G x , fixed in the plane n x and 
carried along with it, always touches a fixed circle C in the plane n, then this same 
condition may be expressed indifferently in either of the forms—(1) a circle C 1 in the 
plane Hj always passes through a fixed point of n ; (2) a point in the plane lij is 
always situate on a fixed circle C in the plane n. But if either of the circles C, G x 
reduce itself to a line, then we have two distinct forms of condition; viz., first, if a 
fixed line in the plane n x always touches a fixed circle G in the plane n, this 
is equivalent to the condition that a fixed line L l in the plane n x always passes 
24—2