TRANSFORMATION OF SURFACES BY BENDING. 



465 



d 3 Id ,2 ds' 1 \ 2 ds' l d \ l d ,, ^ 



and dMt G ° g9) = \dZ [? A? dn^J " ? dV sii^ fe l0Sq ) q dt (1 ° g9) 



2 ds' l Id 



r du sin (p q dudt ^ 0g<? -' , 



two independent values of the same quantity, whence the required conditions may be 

 obtained. 



Substituting in these equations the values of those quantities which occur in the original 

 equations, we obtain 



1 ds I d , / ds . \ 2 ds 1 



9 r^fc' log l pr ^ sin ^ + ;^' cot ^ 



1 1 d$' { d . I , ds . \ 2 ds ) 



= qr' du- {*? l0g IT dJ Sm V + r'Tu COt n' 



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 



A 

 du 



0) 



dh l ° 8 (' 



or 



r du 



= 0. 



(2) 



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 continually in motion with respect to the surface, so that the nature of 

 the polyhedron would need to be altered at every instant. 



60—2 



