682 MOORE. 



the origin, that is through the point of intersection of the fixed planes. 

 The curvature of a geodesic always lies in the normal plane to the 

 surface from which we can conclude that the path curves are geodesies 

 of the surface (43). We will however show this directly. The para- 

 metric equations of the surface are 



•Ti = a cos V, .To = a sin v, 

 ^ ^ xz = b cos V, X4 = b sin v. 



Then 



ds" = a'-dir- + b-dv'- 



This shows that the surface is developable.^^ The geodesies are then the 

 lines given by the relation 



V = Av + B 



which substituted in the differential equations of the path curves we 



find they are satisfied provided A = -^. Hence the path curves 



are the geodesies. Through each point of the surface passes four 

 geodesies which are circles. The planes of two of them are completely 

 perpendicular being parallel respectively to the two fLxed planes. 

 The other two circular geodesies make equal angles with the two 

 preceding, and have their centers at the origin. Their planes con- 

 sequently intersect in a line, that is, lie in a 3-space. The surface 

 can be generated by rotating any one of the path curves by any 

 transformation of the group. Therefore it can be generated by moving 

 a circle of fixed radius and plane parallel to the a-3.r4-plane with center 

 in the a'iX2-plane so that it always cuts a fixed circle lying in a plane 

 parallel to the a-ia-2-plane. It can also be generated by a circle of 

 radius Va^ -f b^ with center at which always cuts a fbced circle of 

 radius Va^ -}- b- ^^^ center at 0. The plane of the variable circle 

 is inclined at a fixed angle to the plane of the fixed circle. 



Equations (43) show that the surface is of order four. Therefore a 

 3-space which cuts it in a circle must cut it again in a curve of order 

 two. Consider first a 3-space which contains one of the circles with 

 center at 0. This will also contain a second one of these circles since 

 they lie by twos in all the 3-spaces containing one of them. If then 

 we pass a sphere through one of these circles for which it is a great 

 circle, it will contain a second one of them which will also be a great 

 circle. Hence such a sphere is tangent to the surface at two diametri- 

 cally opposite points. 



13 See Levi, loc. cit. 



