Vol. 8, 1922 
MATHEMATICS: J. L. SYNGE 
203 
{x^) system for which the equation of the plane surface is x^ = 0, and 
gNN = 1- Since (3.3) and (4.5) hold at all points of the surface, the equa- 
tions (2.3) for the directions of Type III become 
G^^dx, = Bg^^dx, ((7 = 1, . . ., A^-1) 
Gjsfjsi dx]s[ = ^^ATiv dx]\^ 
These equations are satisfied by dx^ = 0 (cr = 1, . . . , N —1), which is 
the direction normal to the plane surface, and thus the theorem is es- 
tablished for Type III. This part of the theorem has also been proved 
by Ricci {Atti R. 1st. Ven. loc. cit.). 
By (3.3) the equations (2.5) for the directions of Type IV become 
t'' t'' G,,s.,s.. Gt,t,,t.t dx, = dg„4oCr ((7 = 1, . . . , iV- 1) (6.2) 
t'' t'' t'' Gs,s.,s.N G,,t.M dx, = eg^^dx^. (6.3) 
Let us consider the surviving terms in the left hand sides of these equa- 
tions, the relations effecting reductions being from (3.3), 
i" = 0 (a= l,...,N-l)- 
from the well known properties of the tensor-components, 
GNN,st = 0, Gst^NN = 0 (s,t = 1, .. ., A/"); 
while from (4.4) we see that any tensor-component vanishes if one and 
only one of its indices is N. In (6.2), ii t = N, then either ti or t2 must 
be N. Therefore either Si or ^2 must be N; therefore must be A^. Hence 
ts = N, and the second tensor vanishes. Therefore there are no surviving 
terms in (6.2) for which t = N, and (6.2) may be written 
A^^dx^ = Bg^^dx^ ((7=1,..., N-l). (6.4) 
In (6.3) either 5i or 52 must be N. Therefore either ti or t2 must be A^* 
therefore either h or t must be N. But if h = N, then S3 = N and the 
term vanishes. Therefore ^3 =t N, and the only surviving term is that 
for which t = N. Thus (6.3) may be WTitten 
BdxN = Bg^^dx^. (6.5) 
But (6.4) and (6.5) are satisfied simultaneously by dx„ = 0 (cr = 1, . . ., 
A^— 1) and thus the theorem is proved. 
In the case N = 2, the directions of Types III and IV become indeter- 
minate, since through any point we can draw a "plane surface" (in this 
case a geodesic curve) so that its normal may have any arbitrarily as- 
signed direction. 
(6.1) 
