292 
MATHEMATICS: L. P. EISENHART 
Hence X, Y, Z are the direction-cosines of the plane (6), and if x, y, z 
are the cartesian coordinates of a surface then X, F, Z and W are 
the tangential coordinates of its transform 2. These four coordinates 
are solutions of the equation 
H — ■ — I — ~ ir" + ^<^ = ^, \^) 
ouov ov ou ou ov 
where 
/= /— 77 1 . dlogv7 dlogV^ 
Vp = vpo-, ^ = —y= ~^r^ — I — — — + 
Vo- OUOV ov ou 
dlogVp dlogVo^ 
(9) 
Consequently the parametric conjugate system on S has equal tangential 
invariants. 
When the same polar transformation is applied to Si, the tangential 
coordinates of 2i, namely Xi, Yi, Zi, Wi, are obtained from (7) on re- 
placing X, y, zhy Xi, yi, Zi respectively. These functions satisfy an equa- 
tion of the form (8), obtained on replacing p andFby pi and Fi, where in 
consequence of (4), 
/ Vp^'i 
p 
and Fi is analogous to F. 
Because of equations (3), the functions Xi, Yi, Zi, Wi are the respec- 
tive transforms of X, Y, Z, W by means of the equations 
(11) 
where 
Wi = e,/^/ly. ' (12) 
Consequently is a solution of (8). From (9) and (10) it follows that 
XiV^ = - Vppl^i. (13) 
Hence equations (11) are equivalent to those of a transformation^ 12. 
We are now in a position to prove the theorem: 
// 2 and Si are so related that for the congruence of lines of intersection of 
corresponding tangent planes x and tti to 2 and 2i respectively the focal 
planes of the congruence are harmonic to x and tti, and the developables 
of the congruence correspond to conjugate systems on 2 and 2i, the latter 
systems have equal tangential invariants; and 2 and 2i are in the relation 
of a transformation Q. For if we apply the polar transformation to the 
surfaces 2 and 2i, the resulting surfaces are related in the manner which 
