Vol,. 8, 1922 
MATHEMATICS: J. L. SYNGE 
199 
or 
:— G = (5=1,..., N), 
OXs 
But these are of the same form as (2.1), writing 6G = 0, and hence the 
type of principal direction defined above is that for which G^, and con- 
sequently G, is stationary. 
Type II: Consider a point (P) and a geodesic passing through it. Along 
this geodesic 
\_()Xs^Xt (ml (^XynA 
This quantity is a function of direction at P, and the principal directions, 
corresponding to stationary values, are given by 
r - f^l dx, = dg,,dx, (s = 1, ...,iV). (2.2) 
Type III: The expression GstXsXi is invariant for any given direction. 
Principal directions, corresponding to stationary values, are given by 
Gst dxt = Bg^tdxt (5 = 1, . . . , A^). (2.3) 
Eisenhart {Proc. N. A. 5., Vol. 8, No. 2, p. 24) has shown that the prin- 
cipal directions of Ricci {loc. cit.) may be expressed in this form. These 
directions may also be reached from other considerations. Any direction 
at a point (P) defines a surface consisting of all geodesies passing through 
P and perpendicular to the given direction. The curvature invariant 
{G) of this surface at P depends only on the given direction. Those 
directions making G stationary are principal directions: they may be 
proved to be identical with those considered above. 
Type IV: The expression 
t"t"t"Gs,s,.s,sG,,,„t/xsX, (2.4) 
is invariant for any given direction. Principal directions, corresponding 
to stationary values, are given by 
g"" g"" Gs,s,.s.s G,„„t„ dx, = dx, (s = 1,...,N). (2.5) 
The four types defined above are not intended to be exhaustive of all 
types of principal directions. Type II, for example, will give principal 
directions if any other invariant function of position is substituted for 
G. Let us suppose that the expression (2.4) has the same value for all 
directions at any given point, but varies from point to point. The prin- 
cipal directions of Type IV are then indeterminate, but (2.4) is an in- 
variant function of position and may therefore be substituted in (2.2) 
to yield principal directions. 
