ROTATIONS IN HYPERSPACE. 693 



to the j:;-space generated by the tangents to the curves at the point 

 in question. If any of the m's vanish the resulting rotation is equiva- 

 lent to a rotation in a space of lower dimensions and therefore we shall 

 assume that none of the m's vanish. 



Equation (62) shows that, for real values of vii, the end of the curva- 

 ture vector will lie inside a ;;-point A, called the curvature p-point 

 determined by the p points in which the extremity of the vector r 

 projects on the p planes Mi. Each point of A will correspond to 

 2^'"^ directions through P. The points in A which correspond to p 

 mutually perpendicular directions through P for a p-po'mt whose 

 center of gravity coincides with the center of gravity of .1. If a 

 linear relation 1,aimi = exists among the m's (62) shows that the 

 end of the corresponding curvature vectors will lie on a closed quadric 

 in p-1 dimensions which touches the faces of A. The foot of the 

 perpendicular dropped from the point P on the space in which A 

 lies corresponds to the directions on the surface satisfying the rela- 

 tions 



(63) nil = =*= '"2 = =■= nis = ....= ± nip 



These curves then, 2""^ in number, are curves of minimum curvature. 

 The first torsion of the path curves is given by the formula 



T = ^numjimi' - m/)(3/..,)x(j/..,) W _^^ 



\dsJ VC-C 



This formula shows that the curves of zero torsion, excluding those 

 corresponding to rotations in a space of less than 2 p dimensions, are 

 those which satisfy relations (63). Hence these curves are plane 

 curves, that is, circles. It is easy to show that the center of these 

 curves is at the origin or at the intersection of the p invariant planes 

 Mi. Other path curves are circles but these belong to rotations which 

 leave one or more of the invariant planes absolutely fixed, that is, are 

 equivalent to rotations in a lower space. 



The differential equations of the path curves are 



dxi dxi dxz dxi 



-— = miX2, -r-= — mi.vi, —- = m^Xi, -7- = — moXs, 



dt dt dt dt 



One set of integrals of these equations is 



010 90I0 01 O O 



Xi- + X2^ = ar, .1-3- + Xi' = ar . . . . xop.i- -f- xip^ = Up- 



