Vol. 8, 1922 
MATHEMATICS: J. L. SYNGE 
201 
O.O.TX^n) system for which the given surface has the equation x^^ = 0, 
and gNN = 1 throughout the manifold (cf. Bianchi, Vol. I, p. 336). 
4. A "plane" surface — the "superficie geodetiche" of Ricci (Atti R. 
Acc. Lincei, Ser. 5, Vol. 12, p. 409) — may be defined as one whose geodesies 
are also geodesies of the containing manifold. lyCt us choose an O.T. 
{^n) system for which a certain plane surface has the equation = 0. 
The line element of the surface is given by 
and the equations of its geodesies are, in the form of (3.2), 
= 0 (4.1) 
Xn = 0; (4.2) 
for such a curve we find, for o- = 1, . . ., A^"— 1, 
= 0, by (4.1); 
also 
^^'j x^xn = x,x,, by (3.1) and (4.2). 
by (3.1). 
2 bxj^ 
From the definition of a plane surface and by (3.2), this latter quantity 
must vanish for all arbitrary directions in the surface. Therefore we 
must have, at all points of the surface, 
= 0 (m,^ =1, ... ,A^-1). (4.3) 
OXn 
If we are given a plane surface and choose a G.O.T.{xj^) system for which 
the equation of the surface is x^ = 0, and g^N = 1> then, at any point of 
the surface, (3.3) and (4.3) hold. Applying these conditions for the 
reduction of (1.1) and (1.2), we find that at any point of the surface 
Gn.m = 0 iv,a,t = 1, ...,7V-1) (4.4) 
Gn. = 0 {cT = 1, ...,N-1) (4.5) 
5. Theorem: The direction of Type I is contained in those of Type II if, 
■and only if, the lines of Type I are geodesic. In order to establish this 
theorem we shall employ an O.TXx^) system for which the surface G = 
constant have the equations x^^ = constant. The principal directions 
•of Type I are then given by 
dx, = 0 ((7=1,. . .,iV-l); (5.1) 
