4o8 Leigh Page, 



= D^lo8:|/- 



where g is the determinant of G^;^. Therefore 



B„„E^ {o-v, TJ- : {rfx, v\ - D • I0-/X, v! - 50-/X, tJ • D \ogVg+D\ \ogVg 



(56) 

 This equation may be interpreted either as defining the char- 

 acter of the curved four dimensional representative surface in 

 homoloidal space of higher dimensions, or as the law of gravi- 

 tation. It is of interest to follow its significance a little further 

 from the first aspect. Let the rectangular coordinates zc^ . . . 

 W5 of five dimensional homoloidal space be so chosen that w^ is 

 perpendicular to this surface at a given point, and that zi\ . . . zv^ 

 have the directions of the lines of curvature at this point. Then 

 if ^1 . . . k^ are the principal curvatures, 



in the neighborhood of the given point, which has been chosen as 

 origin. If this relation is used to eliminate zv^, the linear element 



-|- 2k.^k2ZJ0-^zj02dzju-^dzVo . . . (57) 



As it is an invariant it vanishes when calculated in terms of 

 the o-'s defined by (57) as well as when expressed in terms of h 

 and k. Its value at the origin is particularly easy to determine, 

 for there 



g'"^=g =I, D G . = c;, 



\X.U T VK 



and consequently we have 



Substituting for the polyadics their values 



hh + hh + ^3^'x + hK + KK + JhK = 0. (58) 



The left hand member of this equation measures the curvature 

 of the four dimensional representative surface in much the same 



