no 
ON THE HOMOGRAPHIC TRANSFORMATION OF 
[122 
show that the point (£, y, £, co) lies in the line joining the points (x x> y x , z lf w x ) and 
(x 2 , y 2 , z 2 , w 2 ); and to show that this line touches the surface V = 0, it is only 
necessary to form the equation of the tangent plane at the point (£, y, £, co) of the 
surface in question ; this is 
(x + vy —yz + aw) (g + vy — y% + aw) + ... = 0 ; 
or what is the same thing, 
(x + vy — yz 4- aw) x x + ... = 0, 
which is satisfied by writing (x x , y x , z x , w x ) for (x, y, z, w), that is, the tangent plane of 
the surface contains the point (x x , y x , z x , w x ). We see, therefore, that the line through 
(x X) y x , z X) w x ) and (x 2 , y 2 , z 2 , w 2 ) touches the surface V=0 at the point (f, y, g, co). 
Write now 
- p 
/ ^ 7 / 
a = —jr 7 o = , 
9 9 
r'-Zl v - — 
f f 
y = 
T’ 
/ -c. 
if we derive from the coordinates x 1} y x , z 1} w u by means of these coefficients 
a\ b', c, X', y, v, new coordinates in the same way as x 2 , y 2 , z 2 , w 2 were derived by 
means of the coefficients a, b, c, X, y, v, the coordinates so obtained are — x 2 , — y 2 , — z 2 , — w 2 , 
i.e. we obtain the very same point (x 2 , y 2 , z 2 , w 2 ) by means of the coefficients (a, b, c, X, y, v), 
and by means of the coefficients (a', b', c', X', y, v). Call f', y, w! what £, y, w 
become when the second system of coefficients is substituted for the first; the point 
£', y, £', w' will be a point on the surface V' = 0, where 
V' = (fy* (x 1 + y 2 + z~ + tv 2 ) 
+ (— cy + bz — Xw) 2 + (— az + cx — yw) 2 + (— bx + ay — vw) 2 + (—Xx — yy — vz) 2 ; 
and since 
V + V' = k (x* + y 1 + z 2 + w 2 ), 
and V = 0 intersects the surface x 2 + y 2 + z 2 + w 2 = 0 in four lines, the surface V' = 0 
will also intersect this surface in the same four lines. And it is, moreover, clear that 
the line joining the points (aq, y u z 1} w x ) and (x 2 , y 2 , z 2 , w 2 ) touches the surface V' = 0 
in the point (£', y, to'). We thus arrive at the theorem, that when two points 
of a surface of the second order are so connected that the coordinates of the one 
point are linear functions of the coordinates of the other point, and the transformation 
is a proper one, the line joining the two points touches two surfaces of the second 
order, each of them intersecting the given surface of the second order in the same 
four lines. Any two points so connected may be said to be corresponding points, or 
simply a pair. Suppose the four lines and also a single pair is given, it is not for 
the determination of the other pairs necessary to resort to the two auxiliary surfaces 
of the second order; it is only necessary to consider each point of the surface as 
determined by the two generating lines which pass through it; then considering first