ROTATIONS IN HYPERSPACE. 683 



The plane of one of these circles is ohtalnefl by letting v = u -\- d 

 in equations (44). The equation of this path curve then becomes 



/_,->, .ri = cos w, .To = a sin u 



^ ^^ .1-3 = h cos (v + e), .7-4 = b sin (u + 6) 



and the plane of the curve becomes 



* n ^ ■ n 

 .Ts = - cos d :c\ sm dx^, 



a a 



(46) 



Xi = - cos d X2 + - sm 6xi. 

 a a 



Varying 6 we obtain all circles which form one generation of the sur- 

 face. These circles have no point in common. The second generation 

 can be obtained by putting « = — \i -\- 6. The equations of these 

 circles then is 



. _^ Xi = a cos u, .T2 = a sin u 



^ '^ .T3 = h cos(— u + 6), Xi = h sin (— u + S). 



The circles of (47) do not intersect each other but each one of (47) 

 intersects each one of (45) in two diametrically opposite points. 



The points of intersection are u = — ; — -, u = ; — . If these 



circles are used as parameter curves the equation of the surface 

 becomes 



Xi = a cos (?/ + v), X2 = a sin {u + v), 

 Xs = b cos {u — v), Xi = b sin (m — v). 



From (46) we see that the locus of the planes of these circles is 



Xj^ + X 2^ _ Xz^ + X^ 



a^ ~ b^ ' 



In fact this quadric cone contains both sets of planes. 



The planes of the other two generations of circles lie on the cylinders 



Xi- + X2~ = a^, x^ + Xi = b-. 



The planes on one of these cylinders are parallel to each other and 

 consequently two of them determine a 3-space, that is, the 3-space 

 which passes through one of these planes will contain another of the 

 same cylinder. Then in this second double generation of circles, 



