198 
MATHEMATICS: J. L. SYNGE 
Proc. N. a. S. 
PRINCIPAL DIRECTIONS IN A RIEMANNIAN SPACE 
By J. L. Synge 
De^partment of Mathematics, UnivbRvSity of Toronto 
Communicated, May 23, 1922 
1. A type of principal directions for a Riemannian space of N dimen- 
sions has been defined by Ricci (Atti R. 1st. Veneto, 63, p. 1233). Four 
distinct types are defined in the present paper, of which one is identical 
with that of Ricci, although defined in a manner somewhat more simple. 
We shall adopt the common convention of summation with respect to 
any index occurring twice in a product, except where the index is a capital 
letter. The manifold under consideration is of N dimensions; the small 
Roman indices imply a range or summation from 1 to N, the small Greek 
indices from 1 to N—1. The line element being given by 
we define in the usual manner 
- 1 [:] -s. [:'] + '•'{[:][:] - [:][:]} 
Gns — Gmn,st (1-2) 
G = g"' G„,. 
The word "surface" will be used to denote any (AT— 1) -space immersed 
in the given A/^-space. 
2. Directions defined by invariant relations may be termed principal. 
Any invariant function of direction will, in general, yield such principal 
directions, corresponding to stationary values of the function. The follow- 
ing are types of principal directions: — 
Type I: Consider the family of surfaces, G = constant. Its orthog- 
onal trajectories constitute principal directions; their equations are 
^ = Hs,ic, {s=l,...,N) (2.1) 
where the point denotes differentiation with respect to the arc. Now, 
for any direction, 
bG bG 
— ■ — axs dxt 
^2 ^ bxs dxt . 
gst dxs dxt 
therefore directions making stationary satisfy 
^ ^ dxi = (j>g,i dxt {s = 1, . . N) 
OXs OXt 
