dxt 



SURFACES IN HYPERSPACE. 319 



XMX(«) = 0. 



V ^^'■^ x^r)x{s) _ 2S,X,3X('^ - 22,. 



ts 

 r 



The first and last terras cancel and hence the condition for a geodesic 

 in the notation of the covariant derivatives is 



i:,\uV'^ = 0. (79) 



In terms of the system (ps derived from the X's the condition is, by (64'), 



2.Xt^.X^^> = or S,.^,X<*) = 0. (79') 



The quantity 2(^sX^*' is an invariant which vanishes when X is a 

 system of geodesies. 



32. Curvature; Interpretation of a and y. The moving axis 

 I is tangent to the curves X. The curvature of these curves is ds/ds 

 and from (71) takes the form 



ds ds 



Hence 



c = f=a + 7ii, (80) 



ds 



if 



7 = X<pr\^^\ (81) 



where y is the invariant which vanishes (as has been seen) for geo- 

 desies. The curvature of a surface curve therefore has two components 

 one normal to the surface and equal to the vector invariant ci, one in the 

 surface perpendicular to X and of magnitude y. We have therefore an 

 interpretation of the vector invariant ci, namely, the component of 

 the curvature perpendicular to the surface. We have also an inter- 

 pretation of 7 as the tangential component of the curvature. A geo- 

 desic being a curve which has no tangential component of curvature, 

 the curvature of a geodesic is wholl}^ normal to the surface, i. e., the 

 osculating plane of the geodesic is normal to the surface, no matter xohat 

 ike number of dimensions in which the surface lies. We may conse- 

 quently say that: the vector a is the curvature of the geodesic which is 



