TRANSFORMATION OF SURFACES BY BENDING. Ill 



Taking two consecutive positions of this line, in which the values of t 

 are t and t + St, we may find by the ordinary methods the equation to the 

 shortest line between them, its length, and the co-ordinates of the point in which 

 it intersects the first line. 



Calling the length S, 



and the co-ordinates of the point of intersection are 



x = 2a cos 3 1, y = 2a sin 3 1 , z= c cos 2t. 

 The angle 80 between the consecutive lines is 



The distance So- between consecutive shortest lines is 



and the angle 8$ between these latter lines is 



-ct 



Hence if we suppose , 6, cr, <j>, and t to vanish together, we shall have by 

 integration 



^27^ (1 - C082<) ' 



(T = 



= ^L(1-COS20, 



2Va' + c jV 

 <H-7^=*. 



I ft* _L y>> 



By bending the surface about its generating lines we alter the value of < 

 in any manner without changing , 6, or o-. For instance, making <f> 0, all the 

 generating lines become parallel to the same plane. Let this plane be that of 

 xy, then is the distance of a generating line from that plane. The projections 



