64 
MATHEMATICS: L. P. EISENHART 
For the sake of brevity we put 
T', = a'i + b^ + wj, H = VeG^\ (8) 
The functions ao, bo, Wq , Xq , which determine Co are given by 
ao = - since- Wop(\/g — + V E — 
\ ou ov 
cosw'ze'oP (Vg^-Ve~), 
\ Ou ov / 
d log Wo _ D dt d log Wo _ D" dt 
bu H. du dv H dv 
To=HpWo, (9) 
!^ (Xo- P^o) =-^+lt ^logHp, ^ (Xo+p^o) = ^- + 'tt~^ogHp. 
To E — ou To H — dv 
du dv 
The complementary function to for Co is given by 
to = t-^^^. (10) 
XoWo 
The functions for the surface Co' are analogous to the above. 
Ordinarily the surfaces Si derived from a surface C by transfor- 
mations K are not surfaces C. However, the equations 
— + a/e cos coai — sin co&i+ (^i— /) (cos co^o— sin cobo) 
du I To 
0 
j^cos 0)^1 -h sin coZ^iH- (ti — t) -^^^ (cos o)ao+ sin co6o; 
(11) 
dv ^ L 
are consistent with equations (7) for i = 1, the function 6i so defined 
is a solution of equation (1), and the new surface ^i, given by (6), is 
a surface C, say Ci. In particular we remark that the function ti given 
by (2), when Xi and x are replaced by ti and t respectively, is the com- 
plementary function for Ci. Furthermore, if Xi and x in equations (2) 
are replaced by Xi^ -j- y + Zi^ — t-^ and x'^ -\- -\- z'^— t^ the resulting 
equations are satisfied. 
With the aid of the theorem of permutability of general transforma- 
tions X (cf. Transactions^ loc. cit., p. 406) we show \haXij C, Co and Ci 
are three surfaces in the relation indicated above, a fourth surface Cio can 
be found without quadratures such that the lines joining corresponding 
points Ml, MiQ, on Ci and Cio is a normal congruence. 
Likewise it is found that the surfaces U and 12i normal to the congruences 
of the lines MMq and Mi Mio at the distances t and ti from C and Ci re- 
spectively envelope a two-parameter family of spheres, and the lines of 
