200 
MATHEMATICS: J. L. SYNGE 
PROC. N. A. S. 
3. In order to prove certain theorems concerning the principal directions 
defined above, we shall require particular coordinate systems. A system 
of coordinates will be said to be ''O.T.{x^y' if the parametric lines of 
are the orthogonal trajectories of the family of surfaces x^^ = constant. 
The necessary and sufficient conditions for an O.T.ix^) system are easily 
seen to be 
giv.. = 0 (a- = 1, ...,iV-l). (3.1) 
A special type of O.T.{xn) system is the "G.0.T.(^k;2v)" system for which 
the parametric lines of x^ are geodesies. The equations of a geodesic are 
i fTtfl 
+ < ^ > XmXn = 0 (S = 1, ...,N) 
or 
gst ""t + I "^'^ \0CmXn = ^ = 1, . . . , TV). (3.2) 
[mn~\ ■ • _ 
S J 
The coordinate system being O.T.(^>), the parametric lines of x^^ satisfy 
(3.2) if, and only if, 
(cr=l, ...,A^-1) 
and 
gNN Xn-\-\' 'V \ =0. 
The latter equation is always satisfied for the parametric lines of x^, by 
virtue of the equation 
gNN ^N — 
the former are equivalent to 
0 (o- = 1, ...,7V-1). 
<>gNN 
()X(T 
Hence we have, as necessary and sufficient conditions for a G.O.T. 
(^n) system, 
gN. = 0, ^ = 0 ((T = 1, ...,iV-l). (3.3) 
dx^ 
If we are given an oo i family of surfaces, we can find an O.T.{x^) system 
for which the family is given by x^ = constant. If we are given a single 
surface and draw the congruence of geodesies normal to it, it follows from 
the Calculus of Variations that this is a normal congruence, and that any 
two of the normal surfaces give equal intercepts on all the geodesies. 
Taking these geodesies as parametric lines of x^ and taking x^^ as the dis- 
tance measured along these geodesies from the given surface, we have a 
