66 G. FLEURI. 
angles pom and mop—! of the preceding figure correspond on the 
figure projection two angles pom and mok such that (0 .i pm) = 
(o.ymk) and determining the point v’ by the condition (i7ku') = 
—1 we have the required tangent w’. Analogous constructions 
will give us the other points and tangents. For instance deter- 
mine op’, by the relation (0.1jmp)=(0.1jpp',) anda point @ by 
the relation (i7p',a)=—1, then draw ap. wp isthe tangent to 
the quadric pedal at » and the point p, of intersection of this line 
with op’, gives us the corresponding point », of the second quadric 
positive pedal. I leave it to the reader to imagine the other © 
constructions.” 
This premised, let us consider any transformation whatever of 
the curve C; for instance let us take the ath positive pedal of O, 
then the inverse of the result, then the reciprocal polar of the 
result, then the a'th negative pedal of the result and finally the 
tangential inverse of the result. That is to say let us make the 
operation 
NT mei) Jeo 
We can consider, in this expression, the two letters R and Jas 
associated ; we have only to consider the curve P— instead of 
the curve C to see the truth of this proposition for we have shown 
above that, making on any curve whatever the operation R J or 
making the operation P~1, gives exactly the same result. 
This is true for any number of letters, for let for instance be 
PO ees 
2.€. let us suppose that the succession of operations indicated in 
the first member of this equality is equivalent to the transforma- 
tion S, then as C represents any curve whatever, instead of C we 
can consider the curve P* and the theorem is thus obvious ; 2.e. 
The geometricai transformations considered being represented 
as above, any transformation whatever can be represented by an 
algebraical expression which is associative. 
From this theorem it results at once that 
Pe P> = Patd 
