127] 
133 
127. 
ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE 
OF THE SECOND ORDER INTO ITSELF. 
[From the Philosophical Magazine, vol. vil (1854), pp. 208—212: continuation of 122.] 
I PASS to the improper transformation. Sir W. R. Hamilton has given (in the note, 
p. 723 of his Lectures on Quaternions [Dublin, 1853)] the following theorem:—If there 
be a polygon of 2m sides inscribed in a surface of the second order, and (2m — 1) of 
the sides pass through given points, then will the 2m-th side constantly touch two 
cones circumscribed about the surface of the second order. The relation between the 
extremities of the 2m-th side is that of two points connected by the general improper 
transformation; in other words, if there be on a surface of the second order two 
points such that the line joining them touches two cones circumscribed about the 
surface of the second order, then the two points are as regards the transformation 
in question a pair of corresponding points, or simply a pair. But the relation between 
the two points of a pair may be expressed in a different and much more simple 
form. For greater clearness call the surface of the second order U, and the sections 
along which it is touched by the two cones, 0, 0; the cones themselves may, it is 
clear, be spoken of as the cones 0, 0. And let the two points be P, Q. The line 
PQ touches the two cones, it is therefore the line of intersection of the tangent 
plane through P to the cone 0, and the tangent plane through P to the cone 0. 
Let one of the generating lines through P meet the section 0 in the point A, and 
the other of the generating lines through P meet the section 0 in the point B. 
The tangent planes through P to the cones 0, 0 respectively are nothing else than 
the tangent planes to the surface U at the points A, B respectively. We have there 
fore at these points two generating lines meeting in the point P; the other two