60 G. FLEURI. 
ON CERTAIN GEOMETRICAL OPERATIONS—Parr I. 
By G. Fieuri, Licencié és-sciences mathematiques and 
Licencié és sciences physiques. 
[With Plate I.] 
Communicated by H. C. Russell, B.4., ¢.M.G., F.R.S. 
[Read before the Royal Society of N.S. Wales, June 1, 1892. | 
I purpose to study in the present paper interesting relations 
which exist between a given curve and curves resulting from it by 
certain well known geometrical transformations. In the first 
part, I will show how the geometrical transformations in question 
can be considered as algebraical operations of a peculiar character, 
and I will establish the principal rules for such operations. 
First Part. 
Let us consider in a plane a fixed circle and a curve C. We 
can deduce from C by well known geometrical transformations 
the following curves : 
First, the reciprocal polar of C with regard to the circle. 
Second, the inverse of C—the centre of the circle being taken as 
origin and the square 2? of its radius as power of inversion. 
Third, the tangential inverse of C with regard to the same 
origin and same power of inversion.* 
Fourth, the successive positive pedals of C, the centre of the 
circle being taken as origin. 
* As this transformation is not so well known as the others I give here 
its defiinition :—Let m’ be a tangent to C, draw op perpendicular to m’, 
determine on op a point r such that or. op= ? and draw rt parallel to m. 
The envelope of the straight line rt is what is called the tangential inverse 
of C. For instance, the tangential inverse of a point is a parabola hay- 
ing this point for its focus and for axis the line joining this point to the 
origin. It is obvious that this transformation is reciprocal like that of 
inversion, i.e., two curves being given, if the first one is the tangential 
inverse of the second, this second curve is also the tangential inverse of 
the first one. 
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