TRANSFORMATION OF SURFACES BY BENDING. 



85 



II. 



On the Betiding of Surfaces of Revolution. 



In the cases previously considered, the bending in one part of the surface 

 may take place independently of that in any other part. In the case now 

 before us the bending must be simultaneous over the whole surface, and its 

 nature must be investigated by a different method. 



The position of any point P on a surface of revolution may be deter- 

 mined by the distance PV from the vertex, measured 

 along a generating line, and the angle AVO which 

 the plane of the generating line makes with a fixed 

 plane through the axis. Let PV=s and AVO = 0. 

 Let r be the distance (Pp) of P from the axis; r 

 will be a function of s depending on the form of the 

 generating curve. 



Now consider the small rectangular element of the surface at P. Its length 

 PIZ = 8s, and its breadth PQ = rS0, where r is a function of s. 



If in another surface of revolution r is some other function of s, then the 

 length and breadth of the new element will be 8s and r'86', and if 



r' = fir, and ff = - 6, 

 P- 



and the dimensions of the two elements will be the same. 



Hence the one element may be applied to the other, and the one surface 

 may be applied to the other surface, element to element, by bending it. To 

 effect this, the surface must be divided by cutting it along one of the generating 

 lines, and the parts opened out, or made to overlap, according as p, is greater 

 or less than unity. 



To find the effect of this transformation on the form of the surface we 

 must find the equation to the original form of the generating line in terms of 

 s and r, then putting r' = p.r, the equation between s and r' will give the form 

 of the generating line after bending. 



