82 TRANSFORMATION OF SURFACES BY BENDING. 



I. 



On the Bending of Surfaces generated by the motion of a strain/lit line in space. 



If a straight line can be drawn in any surface, we may suppose that 

 part of the surface which is on one side of the straight line to be fixed, 

 while the other part is turned about the straight line as an axis. 



In this way the surface may be bent about any number of generating lines 

 as axes successively, till the form of every part of the surface is altered. 



The mathematical conditions of this kind of bending may be obtained in 

 the following manner. 



Let the equations of the generating line be expressed so that the constants 

 involved in them are functions of one independent variable u, by the variation of 

 which we pass from one position of the line to another. 



If in the equations of the generating line A a, u = u l , then in the equations 

 of the line Bb we may put u = u i , and from the equations of these lines we 

 may find by the common methods the equations of the shortest line PQ between 

 Aa and Bb, and its length, which we may call 8. We may also find the 

 angle between the directions of A a and Bb, and let this angle be 80. 



In the same way from the equations of 

 Cc, in which u = u s , we may deduce the equa- 

 tio'ns of RS, the shortest line between Bb and 

 Cc, its length S,> and the angle 80.; between 

 the directions of Bb and Cc. We may also 

 find the value of QR, the distance between 

 the points at which PQ and RS cut Bb. 

 Let QR = 8cr, and let the angle between the 

 directions of PQ and RS be 8$. 



Now suppose the part of the surface between the lines Aa and Bb to be 

 fixed, while the part between Bb and Cc is turned round Bb as an axis. The 

 line RS will then revolve round the point R, remaining perpendicular to Bl>, 

 and Cc will still be at the same distance from Bb, and will make the same 

 angle with it. Hence of the four quantities S&,, S0 3 , So- and 8<f>, 8<f> alone will 

 be changed by the process of bending. S<, however, may be varied in a 

 perfectly arbitrary manner, and may even be made to vanish. 



