ROTATIONS IN HYPERSPACE. 689 



magnitude. The vector r will vanish if vii = iiij = 0, i 9^ j or 

 vii = 0, lUj = lUk a 9^ j 9^ A-), or if nil = ±W2 = =^?»3- The first 

 case gives the transformations which leave two of the planes absolutely 

 fixed and the path curves are circles whose center is the projection of 

 the end of r on the absolutely fixed 4-space determined by the two 

 absolutely fixed planes. The second correspond to the rotations 

 leaving one of the planes absolutely fixed and the path curves are 

 circles with center on the fixed plane and radius equal to the length 

 of the perpendicular dropped from the end of r to the fixed plane. 

 The curvature of the path curves for the last case is 



Mv{Myr)^Mr{M,-r) + Mr{Mrr) _ 

 {Mvrr-+{Mrrr-+{Mrry 



since Mi- (Mi-r) is the projection of r on Mi and the sum of the pro- 

 jections of r on three mutually perpendicular planes is equal to r. 

 Also from the definition of the dot product it is evident that the 

 magnitudes of Mi-r and Mi- (Mi-r) are equal. Hence these curves 

 have curvature directed through the origin and are circles with center 

 at the origin. The point on the curvature triangle corresponding to 

 the direction of the tangents to these circles is the end of the vector 



_ MvjMvr) + Mo-jMrr) + M^-(Mrr) 

 {M^->f+{MrrY-\-(Ms-rr~ 



This vector is perpendicular to the plane of the curvature triangle. 

 For two sides of the triangle are 



Mi-{Mi-r) MrjMrr) Mv(Mi-r) Ms-jMyr) 

 {Mvrf {MrrY ' {M.-rf {Mz-rf 



and it is seen at once that the dot product of C with either of these 

 vectors vanishes and hence C is perpendicular to the plane determined 

 by these two vectors. The four directions ?»i = ='=?«2 = =*=W23 

 correspond to the same point in the curvature triangle viz. the foot 

 of the perpendicular dropped from the end of r on the plane of this 

 triangle. These circles are then the path curves of minimum curva- 

 ture. The radius of curvature being the reciprocal of the curvature; 

 The locus of the centers of curvature of all path curves which pass through 

 a given point is the inverse of the curvature triangle u'ith respect to a 



