114 
NOTE ON THE PROBLEM OF PEDAL CURVES. 
[329 
carrying along with it the point S', to roll on the similar and equal fixed curve OP 
symmetrically situate on the other side of the common tangent OM or OM'; then 
when P' coincides with P, the point S' is brought to S", where SNAT'S" is the 
perpendicular from S on the tangent PN or PN', and SN = N'S", that is, SS" = 2SN; 
and the curve generated by S" (that is S'), or say the epicycloid the locus of S', is a 
curve similar to and similarly situate with the pedal curve the locus of N, but of 
twice the linear magnitude of the pedal curve. Or, what is the same thing, if instead 
of the given curve we consider a similar and similarly situated curve of twice the 
linear magnitude (the point S being the centre of similitude), then the epicycloid the 
locus of S' is the pedal curve of the substituted curve in relation to the point S. 
It may be added that, in accordance with a theorem of Dandelin’s, if rays proceeding 
from the point S are reflected at the given curve, then the epicycloid (or pedal) in 
question is the secondary caustic, or an orthogonal trajectory of the reflected rays. 
2, Stone Buildings, W.G., June 3, 1863.