62 G. FLEURI. 
The reciprocal polar of the pedal of a curve C is the tangential 
inverse of this curve. 
Join om, draw 77 perpendicular to om meeting om at 2 and draw 
mp_y parrallel to rz. By definition, the envelope of mp_, is the 
first negative pedal of C but because 72 and mp are antiparallel* 
therefore : 
0. ON — Of. Opi has 
whence the locus of the point 7 is the inverse of C and mp_, is the 
polar of 2, that 1s to say : 
The reciprocal polar of the inverse of a curve C is the first 
negative pedal of this curve. 
From the well known construction of the tangent to the inverse 
of a curve C which consists, being given the point m and the 
tangent m’ of c, in drawing 7a in such a manner that the triangle 
mai be isosceles, we can deduce the following one: Draw op_y 
making angle 2p_j;=angle roi (om must always be bisector of 
angle pop_;) and ia perpendicular to op_y, ta is the tangent. 
Then, as the pedal C is the inverse of its reciprocal polar, we 
can, from this construction, deduce the following one for the 
tangent to the pedal. Draw op, making angle pop, =angle mop, 
then the perpendicular pp, to op, is the tangent to the pedal and 
obviously the locus of p, intersection of pp, and op, is the second 
pedal of C; a construction identical to the preceding one will give 
the tangent to the secund pedal and the point of the third pedal 
coresponding to the point m of C and so on. Making the angles 
Dope — P,10P. — 9), OD — eee =Dq (Opa = VS eee and 
advaWwiIle, Dos, (siDay 2 een ae Daa Duralte ae acre respectively per- 
pendicular to op.,,.0p2, . ». ci, sO0aen eee the locus of the 
point of intersection of the two corresponding lines pq fa+1 and 
Opa+1 v.é. the locus of the point pa+1 is the (a+1)th pedal of C 
and the line pa+1a+2 1s the tangent to this pedal at the point p,_41. 
An analogous construction based on the construction given 
above of the tangent to the inverse and on the fact that the first 
* Or because the two triangles oir and opm are similar. 
