ON CERTAIN GEOMETRICAL OPERATIONS. 61 
Fifth, the successive negative pedals of C with regard to the 
same origin.* 
There exists between the inverse, the reciprocal polar and the 
first positive pedal of a curve C as above defined, a relation first 
discovered by T. A. Hirst which can be expressed as follows : 
The inverse of the’ reciprocal polar of a curve C is the pedal of 
this curve ; or 
The inverse of the pedal of a curve C is the reciprocal polar of 
this curve. 
This relation is not the only one to be found ; in fact there are, 
as T. A. Hirst himself has pointed out, an infinity of other similar 
relations which can be easily seen by examining the figure relative 
to the construction of the points of the transformed curve corres- 
ponding to a point of C. As these relations will be useful to us 
in what follows, I will show, as succintly as possible, how the 
points and tangents of the transformed curves corresponding to a 
point and tangent of C can be constructed. Let m be a point of 
C and m’ the tangent at this point ; draw op at right angles to m’ 
—meeting m’ at p; the locus of p is by definition the pedal of C. 
Determine r by the relation or.op=A?; ris the pole of m and 
therefore the locus of r is the reciprocal polar of C and the locus 
of p is the inverse of the locus of 7, that is to say: 
- The inverse of the reciprocal polar of a curve C is the pedal of 
this curve. 
Draw rt parallel to m’, the envelope of rt is by definition the 
tangential inverse of C but rt is the polar of » and therefore the 
envelope of rt is also the reciprocal polar of 7p 1.e. 
* Mathematicians differ, with regard to definition of negative pedals, 
as to the point of departure. Salmon (Higher plane curves) starts with 
the first positive pedal in such a manner that the first negative pedal of 
a curve C is the curve itself, and this definition is the most generally 
adopted. However, in view of obtaining a greater symmetry in my 
notations, I have taken—as Clifford and cther geometers have done— 
the curve itself as a point of departure. Consequently what is called 
first negative pedal in this paper corresponds to the second negative 
pedal of Salmon and so on, the order of the negative pedals from Salmon’s 
definition being greater by unity than the order of the corresponding 
pedals as considered in this paper. 
