ON CERTAIN GEOMETRICAL OPERATIONS. 65 
result as it is the case for the products of factors. However one 
of the most important properties of the product of factors: the 
associativity* subsists for our expressions. 
Previous to establishing this fact, I will first generalize the 
meaning of the notations I have adopted by taking a projection 
of the figure we have considered. 
The circle becomes a conic, the straight line at infinity the line 
47, the asymptotes of the circle the lines 02 and oj tangents to 
the conic and the circular points at infinity the points of contact 
4 andj of the tangents 012, 07. 
& will then denote the operation of polar reciprocity with regard 
to the conic. 
I the quadric inversion, or as it is sometimes called the triangular 
inversion. 
T the tangential quadric inversion. 
P*% an operation analogous to the ath pedal and which for want 
of any other name I will call ath quadric pedal—positive or 
negative according toa being positive or negative. 
In order to facilitate to the reader the consideration of these 
operations, I will indicate—as I have previously done with regard 
to the former operations considered—the constructions of the 
points of the transformed curves and tangents thereat correspond- 
ing to a point of the primitive curve C and tangent thereat. 
Let us start with the construction of the tangent to the quadric 
inverse. 
Let / be the harmonic conjugate, with regard to the conic, 
of a point mofC. The locus of / is, by definition, the quadric 
inverse of C. As the anharmonic ratio of the pencil formed by 
the sides of an angle @ and the straight lines joining its vertex. to 
the circular points at infinity is equal to e* 6 (where 2 = ./_j 
eis the well known series) that is to say depends only on the 
value 6 of the angle and on constant quantities, to the two equal 
* In a product of factors, the product of any number of consecutive 
factors can be considered as effectuated. 
E—June 1, 1892. 
