ON CERTAIN GEOMETRICAL OPERATIONS. 63 
negative peaal of C is the reciprocal polar of its inverse will fur- 
nish us with tangents and points of successive negative pedals 
corresponding to tangent m’ and point m of C: Make the angles 
pom — Mop 1 — p21 0p-g =... + 5: = Pa OP—(a+1) +--+. . and 
kan ep 1, 1 Po... -- DAG Ne a rare respectively 
perpendicular to om, op_,..... Cente ROT ; the envelope 
of p_a P—(a+1) is the ath negative pedal of C and the point p_, 
of intersection of p_g p—(a+1) With op_, is the point of tangence 
of this pedal on p_, p_a+1). * 
The construction of the point of the tangential inverse at which 
the line ré is tangent to this curve is deduced from the fact that 
the tangential inverse of C is the reciprocal polar of its pedal by 
constructions analogous to the preceding ones. Make angle pop, 
=angle mop, then the point ¢ of intersection of op, with rtis the 
point of contact sought for. 
A simple glance at the figure will now enable us to write the 
relations of which we have spoken. I will not insist on this point. 
But in order to represent these relations in a convenient manner, 
I will adopt the following notations : 
Let & denote the operation which consists in taking the 
reciprocal polar of a curve with regard to the fixed circle. 
Let J denote the operation of inversion as above defined. 
Let 7’ denote the operation of tangential inversion as above 
defined. 
* It is curious to notice that the points m, p1, po... . pa... p—1, 
0 ae are situated ona logarithmic spiral whose equa- 
tion in polar co-ordinates p and w is, with regard to o as pole and om as 
polar axis: Ww 
P= X cos ack 
where « represents angle mop and A the length om. Forw=a«,2a«,3 
eee ax ..... wehave the different points pj,p2, ps... -.. pa 
of the successive positive pedals and forw=- ~,-2a,3a«..... 
= 0 oe We have the points p_), p29, p-3 -.... [1S A ee 
of the successive negative pedals, all these points corresponding to point 
m of C; for w=o the spiral passes through point m itself. 
