132 Proceedings of the Royal Society of Edinburgh. [Sess. 
rotating round 0 2 , with R on 0 1 P V If R, P v P 2 , . . . P n _ x lie on curves 
C r , C lp , . . . C pn _ v the locus of Q is a curve of order rppp 2 . . . p n _fn + 1). 
The demonstration is similar to that of Prop. VI. 
Cor. 1. The point 0 2 is multiple on the curve of order 
r Pi p 2 . . . p n _ji. 
§ 46. Prop. VIII. 
Generalisation of Prop. XXIV of Part I. 
Consider two serrate angles 
O^P, . . . P m P 
and 
0 2 Q 2 Q 2 . . . QnQ 
in which P X P 2 • . . P m He on curves of orders p v p 2 , . . . p m , and Q X Q 2 
. . . Q n on curves of orders q v q 2 , . . . q n respectively. If P m P and Q n Q 
intersect on a curve C r the intersection of 0 1 P 1 and 0 2 Q 1 generates a curve 
of order 
r(n + m + 2)11^11^. 
SECTION III. 
Maclaurin’s Theory of Pedals. 
§ 47. An intelligent perusal of the preceding shows that Maclaurin 
would inevitably have been led to the pedal transformation of curves,* 
which he now discusses very thoroughly in general terms, along with its 
application to conic sections and other familiar curves. 
He gives no name to the transformation, and the term pedal ( podaire ) 
was introduced by the geometers of the nineteenth century. 
Almost the only nomenclature he introduces, the “ Radial Equation ” for 
the p — r equation to a curve (p. 96), has been quite overlooked and 
adapted to another purpose by Tucker. 
Definition. 
The definition is the usual one. 
* But compare § 2 of Part I. 
