TRANSFORMATION OF SURFACES BY BENDING. 107 



Substituting in these equations the values of those quantities which occur 

 in the original equations, we obtain 



1 ds f d . I ds . ,\ 2ds' .} 

 Q--J- -T-, log (pr j- sin <b )+ - j , cot <& > 

 * r du {du = \f du v / r du r J 



1 1 ds' ( d . / ,ds ,\ 2 ds ,} 

 = > -J-, { -j, log pr -j, sui <b ) + -j- cot (4 >- , 

 q r du \du = \* du Y J r du r j ' 



which is the condition which must hold at every instant during the process of 

 bending for the lines about which the bending takes place at that instant. 

 When the bending is such that the position of the lines of bending on the 

 surface alters at every instant, this is the only condition which is required. 

 It is therefore called the condition of Instantaneous lines of bending. 



17. To find the condition of Permanent lines of bending. 



Since q changes with the time, the equation of last article will not be 

 satisfied for any finite time unless both sides are separately equal to zero. In 

 that case we have the two conditions 



d , / ds ,\ 2 ds' 



or - 3- = 0. 

 r du 



d , I ,ds . \ 2 ds 



Q 



or -, -T-, = 0. 

 r du 



If the lines of bending satisfy these conditions, a finite amount of bending 

 may take place without changing the position of the system on the surface. 

 Such lines are therefore called Permanent lines of bending. 



The only case in which the phenomena of bending may be exhibited by 

 means of the polyhedron with quadrilateral facets is that in which permanent 

 lines of bending are chosen as the boundaries of the facets. In all other cases 

 the bending takes place about an instantaneous system of lines which is con- 

 tinually in motion with respect to the surface, so that the nature of the poly- 

 hedron would need to be altered at every instant. 



142 



