4 
. 7 
. 
\ 
Let P“(a being a whole number) denote the operation which 
consists in taking the ath positive pedal with regard to the centre 
of the fixed circle. 
64 G. FLEURI. 
Let P— denote the analagous operation of the negative pedal. 
Then, considering the curve C as a unit of a certain kind, we 
can express conventionally the relations above mentioned by the 
following algebraical equalities : 
I pede 
st | Fpo ep 
Ind- hha 
3rd RT =P-! 
where J R means the inverse of the reciprocal polar of C 
where P means the first positive pedal of C . 
where & P means the reciprocal polar of the first positive pedal 
of C and so on. 
With the aid of the preceding notations, I give in the following 
table all the relations which exist between two of the transfor- 
mations i, ly 4) and em 
Ji = Jig Ie Mig ie 
Ig sie JB = I TT see 
Sis Ss iP? Ma fe we 
lis Je I M2 its 2 ae 
| ee ee Pee 
hee Dea) fs 
Je ilies 1k PATa Te 
PoP =P? Vie Spe || 
Peet al yi aa es 
Note.—All these relations are not distinct ; for instance, we 
already know that J # = Pand J P= Fare two relations resulting 
from each other. Ina similar manner the relations R J =P —1 
and P—1[ = & are equivalent to the two former relations, etc., ete. 
The expressions such as Jh, JP, RP, RI and so on bear a ; 
striking resemblance with the ordinary algebraical products of i 
factors but the former expressions are not commutative 2.e. the | 
order of the letters cannot be inverted without any change in the 
