134 
ON THE HOMOGRAPHIC TRANSFORMATION OF A SURFACE 
[127 
generating lines at the points A, B meet in like manner in the point Q. Thus P, 
Q are opposite angles of a skew quadrangle formed by four generating lines (or, what 
is the same thing, lying upon the surface of the second order), and having its other 
two angles, one of them on the section 6 and the other on the section <£; and if 
we consider the side PA as belonging determinately to one or the other of the two 
systems of generating lines, then when P is given, the corresponding point Q is, it 
is clear, completely determined. What precedes may be recapitulated in the statement, 
that in the improper transformation of a surface of the second order into itself, we 
have, as corresponding points, the opposite angles of a skew quadrangle lying upon 
the surface, and having the other two opposite angles upon given plane sections of 
the surface. I may add, that attending only to the sections through the points of 
intersection of 6, </>, if the point P be situate anywhere in one of these sections, 
the point Q will be always situate in the other of these sections, i.e. the sections 
correspond to each other in pairs; in particular, the sections 6, <£ are corresponding 
sections, so also are the sections ©, <3> (each of them two generating lines) made by 
tangent planes of the surface. Any three pairs of sections form an involution; the 
two sections which are the sibiconjugates of the involution are of course such, that, 
if the point P be situate in either of these sections, the corresponding point Q will 
be situate in the same section. It may be noticed that when the two sections 0, </> 
coincide, the line joining the corresponding points passes through a fixed point, viz. 
the pole of the plane of the coincident sections; in fact the lines PQ and AB are 
in every case reciprocal polars, and in the present case the line AB lies in a fixed 
plane, viz. the plane of the coincident sections, the line PQ passes therefore through 
the pole of this plane. This agrees with the remarks made in the first part of the 
present paper. 
The analytical investigation in the case where the surface of the second order 
is represented under the form xy — zw = 0 is so simple, that it is, I think, worth 
while to reproduce it here, although for several reasons I prefer exhibiting the final 
result in relation to the form x 2 + y 2 + z 2 + w 2 = 0 of the equation of the surface of 
the second order. I consider then the surface xy — zw — 0, and I take (a, /3, 7, 8), 
(a', /3', 7', S') for the coordinates of the poles of the two sections 6, </>, and also 
(x X) y\> z x , «0, (x 2 , y 2 , z 2 , w 2 ) as the coordinates of the points P, Q. We have of course 
x x y x — z x w x = 0, x 2 y 2 — z 2 w 2 = 0. The generating lines through P are obtained by com 
bining the equation xy — zw = 0 of the surface with the equation xy x + yx x — zw x — wz x = 0 
of the tangent plane at P. Eliminating x from these equations, and replacing in the 
result x x by its value > we have the equation 
(yz 1 - zy x ) (yw x - wy x ) = 0. 
We may if we please take yz x —zy x = 0, xy 2 -\-yx x —zw x —wz x = Q as the equations of 
the line PA ; this leads to 
yz x - zy x = 0, ) yw 2 - wy 2 = 0, 
xy x + yx x — zw x — wz x = 0, j xy 2 + yx 2 — zw 2 — wz 2 — 0,