[328 
329] 
113 
instruction for the 
)f the scroll; for 
>h Kk. By what 
node at 0, and 
moves up to K, 
finite, hence the 
mstruetion shows, 
other two inter- 
the feet of the 
of them at one 
llel—and besides 
he projections of 
cubics will thus 
329. 
NOTE ON THE PROBLEM OF PEDAL CURVES. 
[From the Philosophical Magazine, vol. xxvi. (1863), pp. 20, 21.] 
It is not, so far as I am aware, generally known that the problem of pedal 
curves (Steiner’s Fusspuncten-Gurve) was considered by Maclaurin in the Geometria 
Organica, 1720. He appears to have been led to it through an idea such as 
Sir W. R. Hamilton’s Hodograph, or at any rate with a view to a dynamical appli 
cation, for he remarks, p. 95, “ Cum vero geometria quse curvas ad datum centrum 
relatas contemplatur in philosophia naturali ad motus corporum et vires evolvendas 
facilius applicari possit,... hac sectione considerabimus curvas tanquam ad punctum 
quodvis datum relatas ex quo ad omnia circumferentia; puncta radii undique educuntur, 
et simul perpendicula in illorum punctorum tangentes demittuntur, et rationem radii 
ad perpendiculum tanquam curvae characterem usurpabimus.” And accordingly, Props. 
IX. to XII., he considers the problem: Given a point S in the plane of a given 
curve, to find the locus of the intersection of a tangent of the curve by the per 
pendicular let fall upon it from the point S; with some special cases, and deductions 
from it. In particular if the given curve be a circle, the locus in question (or pedal 
curve) is a curve of the fourth order having a double point S; viz. if S be inside the 
circle, this is a conjugate or isolated point; but if outside, a double point with two 
real branches: if S be on the circle, then instead of the double point we have a 
cusp: it is shown that in each case the pedal curve is in fact an epicycloid. If the 
given curve be a parabola, then the locus or pedal curve is a curve of the third order, 
viz. a defective hyperbola having a double point at S, and with its single asymptote 
perpendicular to the axis of the parabola: some particular cases are specially noticed. 
If the curve be an ellipse or hyperbola, then, as in the case of the circle, the locus 
or pedal curve is a curve of the fourth order having a double point at S. And it is 
moreover shown, Prop. XII., that for any given curve whatever the locus or pedal curve 
is, in a generalized sense of the term, an epicycloid. This is in fact seen very easily by 
a mere inspection of the figure. Imagine the curve O'P', rigidly connected with and 
C. V. 
15