294 
MATHEMATICS: L. P. EISENHART 
it is seen that the coordinates of this point are —d/a, —e/b, —f/c. Con- 
versely, if the Hnes joining corresponding points on two surfaces Z and 
Si, in the relation of a transformation ^2, meet in a point M, the surfaces 
5 and 6*1 arising from 2 and Si by a polar transformation are so placed 
that the lines of intersection of corresponding tangent planes to S and 
Si lie in a plane, the polar plane of M. When in particular, the funda- 
mental quadric of the transformation is chosen so that M is the pole 
of the plane of infinity (which can always be done) 5 and Si are associate 
surfaces. 
When di is a constant, it follows from (12) that Wi is equal to l/\/(r 
to within a constant multiplier, which is unessential, as is evident 
from (11) and (13). From (7) we have 
W + ^X + ^Y + lz^-U'- + ^i+^--X (14) 
a b c Vo- \a b c / 
We choose the quadric so that the coefhcient in (14) of l/\/(Tis not 
equal to zero. Hence Wi is a homogeneous Hnear function of X, Y, Z 
and W, when the lines joining corresponding points on 2 and 2i are 
concurrent. Conversely, suppose that Wi is a homogeneous linear 
function of the form of the left-hand member of (14). Apply to 2 the 
polar transformation with respect to the quadric (5) and let 5 be the 
transform of 2. Take an associate surface of S, say 5*1. When now 
the transformation with respect to the quadric (5) is applied to Si, 
we get a surface 2i in the relation of a transformation Q to 2, the func- 
tion Wi differing by a constant factor at most from the left-hand member 
of (14). Combining this result with the observations made in the 
preceding paragraph, we have the theorem: 
When the function Wi determining a transformation ^ of a surface 2 
is equal to W plus a homogeneous linear function of the direction-cosines 
of the normal to 2, ths lines joining corresponding points on 2 and its 
transform 2i are concurrent; and conversely. 
If in (5) we put c = 0, we have in place of (14) the equation Z = f/y/~of. 
Consequently Wi differs from Z by a constant factor at most, which is 
unessential. Since the z coordinate of the plane at infinity with respect 
to the quadric (5) is infinite, the lines joining corresponding points on 
2 and 2i are parallel. By reasoning analogous to the preceding we 
arrive at the theorem: 
When the function Wi determining a transformation U of a surface 2 
is a homogeneous linear function of the direction-cosines of the normal 
to 2, the lines joining the points on 2 and its transform 2i are parallel; 
and conversely. 
