690 ' INIOORE. 



twit sphere loith center at the point in question and therefore is a sphere 

 with r for a diameter. The centers of curvature of the real curves lie on 

 an octant of this sphere. In four dimensions we found that the path 

 curves corresponding to the center of the curvature segment were 

 orthogonal but here the path curves corresponding to the median 

 center of the curvature triangle do not have this property. 



We saw that the locus of the end of the curvature vector for direc- 

 tions through a given point which satisfy a linear relation, that is 

 curves tangent to the same plane, was a conic. This conic may 

 degenerate into the sides of the curvature triangle counted twice. 

 The directions corresponding to the points of one of these segments 

 are all perpendicular to the direction corresponding to the opposite 

 vertex. To a general point in the curvature triangle correspond four 

 directions through the point but to a general point on one of the sides 

 correspond two directions through the point and to a vertex of the 

 triangle corresponds just one direction. A line in the plane of the 

 curvature triangle is defined by the linear relation 'Lairn^ = 0, and 

 this substituted in (48) shows that the corresponding directions 

 through the point generate a quadric cone. In particular if one of the 

 coefficients, oi say, is zero and the other two have opposite signs then 

 the quadratic relation factors into two linear relations, each of which 

 corresponds to a plane of directions through the point. From which 

 we see that a linear relation involving only two of the /?i's gives a 

 plane of directions whose curvature segment passes through a vertex 

 of the curvature triangle. Two perpendicular directions correspond 

 to the ends of the segment and from the fundamental configuration 

 for the curvature of three mutually perpendicular directions it is at 

 once seen that the curvature of the path curve perpendicular to this 

 plane of directions will cut a side of the curvature triangle. The 

 configuration can be shown by a simple figure. Let ABC be the 

 curvature triangle, AT) \s the curvature segment corresponding to a 

 plane of directions through the point depending on but two of the ?/i's. 

 Let H be the median center of the curvature triangle and G the middle 

 of the curvature segment. The point corresponding to the direction 

 perpendicular to the givea plane i.e. to the directions corresponding 

 to the segment AD must be such that the center of the triangle ADF 

 is H. It is at once evident that F must be on BC and such that BE = 

 EF. This together with a^ = or Os = are the only cases in which 

 ^aiMi^ = can be factored into two linear relations. If D coincides 

 with E, G will coincide with H. 



A line which does not pass through a vertex of ABC will contain an 



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