MATHEMATICS: L. P. EISENHART 
291 
Whenever any two surfaces are in a one-to-one correspondence such 
that the developables of the congruence of lines joining corresponding 
points cut the surfaces in conjugate systems, the Unes of intersection 
of corresponding tangent planes to these surfaces form a congruence 
G' whose developables correspond to the developables of the former 
congruence.^ Evidently two surfaces in the relation of a transforma- 
tion K possess this property, but it is not a characteristic property. 
When any surface S is subjected to a polar transformation with re- 
spect to a quadric, points and tangent planes are transformed into tan- 
gent planes and points respectively of the new surface 2. Since straight 
Hues go into straight lines, it is readily shown that conjugate directions 
on 5 go into conjugate directions on S. Also a congruence of lines is 
transformed into a congruence of Hues, and the developable surfaces of 
the two congruences correspond; furthermore to the focal points on a 
line correspond the focal planes through the corresponding line of the 
other congruence. 
We consider the effect of applying a polar transformation to two sur- 
faces S and 5i in the relation of a transformation K. If the new surfaces 
be denoted by 2 and 2i, the lines joining corresponding points M and Mi 
on these surfaces form a congruence G' whose developables meet 2 and 
2i in conjugate systems, and the tangent planes to 2 and 2i meet in the 
lines of a congruence G whose developables correspond to the develop- 
ables of the congruence G'; moreover, the focal planes of the congruence 
G are harmonic to the tangent planes to 2 and 2i. These properties 
of the surfaces 2 and 2i are possessed likewise by a pair of surfaces in 
the relation of a transformation^ Q. It is our purpose to show that 
2 and 2i are in the relation of a transformation 12, and that the proper- 
ties just mentioned are characteristic of transformations 12. 
The equation of any quadric may be put in the form 
The equation of the polar plane of the point (x, y, z) with respect to 
this quadric is 
ax^ -j- by^ cz^ 2 dx -\- 2 ey 2 fz -\- g = 0. 
(5) 
Xx' -f vy 4- Zz' = w. 
where x', y\ z' are the current rectangular coordinates and 
(6) 
ax-\-d 
Y = 
dx + ey -\- fz-\-g 
a=(ax-\- dy + {by + ey + (cz +J)\ 
(7) 
