202 MATHEMATICS: J. L. SYNGE Proc. N. A. S. 
those of Type II by 
{px^oxi (nt) Ox^J 
^1 dxt = OgNtdxt. 
{px^oxt { m } ox^j 
Applying the conditions, 
{a = 1, 
(5.2> 
ox„ 
to (5.2), we obtain 
- I f I ^ dxt = Bg^^dx, ((T = 1, 
—5- dx^ - < _ > r — dxt = BgNNdx^. 
It is easily seen that (3.1) imply g^" = 0 ((7 = 1 
these equations we find 
2 ox„ 2 
1 NN ^gar 
dxt 
1 
2 d:Ji:A7 
Hence (5.3) become 
dx^ + 
1 
bgi 
\ (5.3) 
. , A/" — 1) ; employing- 
dx, {a = 1, ...,Ar-l) 
NN ^^NN 
bx^ 
dx^. 
1 
NN ^G^ / 
dx^ 
-2' 
bXN \ 
bXN 
/b'G 
- I g^^ 
bG i 
^gNN 
V>xn 
2^ 
dx^ 
bXN J 
= eg,,dx, {(T = 1, ...,Ar-i) 
dXr = dgNN dx^. 
(5.4) 
Since we hypothesize that the Type I direction is determinate, 
bG 
bXN 
+ 0; 
therefore (5.4) are satisfied by (5.1) if, and only if, 
INN 
bx. 
= 0 
((7 = 1, AT-l). 
But, the coordinate system being O.T.(ii[;^), these conditions are necessary 
and sufficient that the parametric lines of x^ shall be geodesic, by (3.3). 
Therefore the theorem is established. 
6. Theorem: If there exists a plane surface, its normal direction is a 
principal direction of both Type III and Type IV. Let us employ a G.O.T. 
