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III. Note on the Problem of Pedal Curves. 

 By A. Cayley, Esq.* 



IT is not, so far as I am aware, generally known that the 

 problem of pedal curves (Steiner's Fusspuncten- Curve) 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 dyna- 

 mical application, for he remarks, p. 95, " Cum vero geometria 

 quse curvas ad datum centrum relatas contemplatur in philo- 

 sophia naturali ad motus corporum et vires evolvendas facilius 

 applicari possit, . . . hac sectione considerabimus curvas tanquam 

 ad punctum quodvis datum relatas ex quo ad omnia circum- 

 ferentise puncta radii undique educuntur, et simul perpendicula 

 in illorum punctorum tangentes demittuntur, et rationem radii 

 ad perpendiculum tanquam curvse characterem usurpabimus." 

 And accordingly, Props. IX. to XII., he considers the problem, 

 viz., 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 at 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 (V P', rigidly connected with and carrying along with 

 it the point $', to roll on the similar and equal fixed curve O P 

 symmetrically situate on the other side of the common tangent 

 O M or OM'j then when P' coincides with P, the point S' is 

 brought to S", where SNN' S" is the perpendicular from S on 

 the tangent P N or P N', and S N = N' S", that is, S S"=2S N ; 

 and the curve generated by S" (that is S ; ), or say the epicycloid 



* Communicated by the Author. 



