ROTATIONS IN HYPERSPACE. 079 



path curves through a point, given by the transformations for which 

 III] = 0, m-2 = 0, lui = IU2, )»i = —))h2 form a harmonic pencil. 



From (42) we see that if /• is held fixed and nii and m^ are allowed to 

 varv the end of the curvature vector traces out the line joining the 



ends of the vectors i^i^ and ^W^) ^„, ,^ ,^ ^^^„ ,,^, ,^^ 

 real values of nii and mo the only points obtained are those on the 



segment joining the ends of these two vectors and that the ratios — 



m2 



and give the same point. Hence each point of the segment is 



m2 



counted twice. We will call this the curvature segment. Two direc- 



dr dr' 



tions - = (niiMi + moMo) • r and — = (mi'^Vi + m^'^h) • r are per- 

 dt dt 



pendicular if they satisfy the relation. 



-•— = = (w,J/i-r + m2Mi-r)-{mi'Mvr + m.'Mrr) 

 dt dt 



= mimi'iMi-ry + momo^Mo-ry. 

 Hence 



^2' _ _ (J/i-r)- mi 

 mi {Mo ■ r)- mo 



Two perpendicular directions are then miMi-r + moMo-r and 

 mo{M.2-r)-{Mi-r) + mi{Mi-r)-{M2-r). The curvature for these two 

 directions is 



ci = [m,UIr(Mvr) + mo^Mo,- Oh- r)](j^' 



02 = [mi^KMo ■ r) • (J/2 • r)]2J/i • (Mi ■ r) 



+ mmii ■ r) ■ (Ml ■ r)YMo ■ (M2 ■ r)](j^^ 

 dt\ 1 



dsJi mi^Mi • rf + »io-( J/2 • rY 

 dt\ 1 



dsj2 ~ mo2[(J/o • r) • (J/2 • /•)]-( J/i • /•)- + m{-[(Mi ■ r) ■ (Mi ■ r^Mo • r)'- 



_ 1 



~ [(Mi-rnM2-rin [mHMi-rf + »;2-(J/2-r)2 



Substituting these values of fyj and f y j in the expressions for Ci 



