232 Mr. A. W. Flux on the 



If we take a plane 



x sin 0—2z + \(z— z x eot 2 0) = O 



passing through the line (vii.), it cuts the surface in the pair 

 of straight lines 



M =+\/ 



x — V 



2 sin 2 0-\ 



It cuts the axis of z where 



Denoting this value of z by z , we see that 



£= + sec0 A /-=^- 

 x ~ V z ± + z G 



gives the pair of lines in which the above plane meets the 



surface. 



It appears, therefore, that the surface is generated by pairs 



of lines intersecting the axis of z between the origin and 



z=—z h and also intersecting the line (vii.). If we denote 



the origin by P (fig. 5), z=—z x by Q, and z=—z by M, 



Fig. 5. 



and also the line (vii.) by RN, R being its intersection with 

 the plane xz, and N its intersection with one of the above 

 pair of lines, 



RN 



2z 1 cos 9 , a /PM 



. + sec 0\J - 



= + 



sin 3 6> 

 2 cos (9 



MQ 



sin 3 d 

 giving the law of the generators 



PQ 



PM 

 MQ' 



