680 MOORE. 



and c-z we have 



That is the sum of the curvature vectors for two perpendicular direc- 

 tions through the point is independent of the pair of directions taken 

 and is equal to the sum of the curvatures of the path curves of the 



rotations mi = and ???2 = smce —j^ — rr- and ——rz — -— are 



the curvatures in these two directions. It is also to be noted that 

 the directions for which the curvatures are the same are harmonically 

 separated by the directions vh = 0, vh = 0. The directions which 

 have curvature equal to h are perpendicular to each other and hence 

 bisect the angle between m\ = and m^ = 0. 



Since the length of the radius of curvature is the reciprocal of the 

 scalar curvature and its direction coincides with C the locus of the 

 centers of curvature is the inverse of the curvature segment with 

 respect to the unit circle with center at the extremity of r. Hence; 

 The locus of the centers of curvature of all the path curves of this group 

 which pass through a given poi^it is a circle of ichich the diameter is the 

 line joining the origin to the point in question. For real directions 

 through the point the centers of curvature lie on a quadrant of this circle. 

 From this it is evident that the curves with minimum curvature are 

 in the directions 7»i = mo and ?»i = —vio and hence for these direc- 

 tions the curvature is perpendicular to the curvature segment. This 

 can be seen also directly from (42). For the curvature of these 

 cur^'es being in the direction of r and the curvature segment being 



Mr(M rr) Mr{M,-r) . 



— 7TT — :::^ rr;^ — ^7— we have 



{Mvrf {Mo-rf 



~Mv{Mvr) M2-{Mrr) 



{Mvrf {Mrrf 



= 0. 



Since ilf 1 • (il/i • r) is the projection of r on M\ and has the same length 

 as Mi-r the product r-[Mi:{Mi-r)] is then the length of the projec- 

 tion of r on ilf 1 multiplied by the length of r. Hence the first term of 

 the product reduces to unity and likewise for the second term and 

 hence the whole product vanishes. 



The curvature vectors of the path curves lie in a plane determined 

 by Mi'{Mi-r) and M^-iMi-r). But Mw is a vector in Mi perpen- 

 dicular to r and Mx-{Mvr) is the projection of r on Mi hence these 



