TRANSFORMATION OF SURFACES BY BENDING. 449 



Now consider the small rectangular element of the surface at P. Its length PR<=Ss, 

 and its breadth PQ=r$6, where r is a function of s. 



If in another surface of revolution / is some other function of s, then the length and 

 breadth of the new element will be Ss and r'&6', and if 



r = fir, and 6' = - 0, 



r'$e = rd9, 

 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 n 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' = fir, 



the equation between s and r' will give the form of the generating line after bending. 



dr . l 



When ft is greater than l it may happen that for some values of s, — is greater than — . 



In this case 



dr dr . , 



— = jul — is greater than 1 ; 



ds ds 



a result which indicates that the curve becomes impossible for such values of s and u, 



The transformation is therefore impossible for the corresponding part of the surface. If, 

 however, that portion of the original surface be removed, the remainder may be subjected to 

 the required transformation. 



The theory of bending when applied to the case of surfaces of revolution presents no 

 geometrical difficulty, and little variety ; but when we pass to the consideration of surfaces of a 

 more general kind, we discover the insufficiency of the methods hitherto employed, by the 

 vagueness of our ideas with respect to the nature of bending in such cases. In the former 

 case the bending is of one kind only, and depends on the variation of one variable ; but the 

 surfaces we have now to consider may be bent in an infinite variety of ways, depending on the 

 variation of three variables, of which we do not yet know the nature or interdependence. 



We have therefore to discover some method sufficiently general to be applicable to every 

 possible case, and yet so definite as to limit each particular case to one kind of bending easily 

 understood. 



The method adopted in the following investigations is deduced from the consideration of 

 the surface as the limit of the inscribed polyhedron, when the size of the sides is indefinitely 

 diminished, and their number indefinitely increased. 



A method is then described by which such a polyhedron may be inscribed in any surface 

 so that all the sides shall be triangles, and all the solid angles composed of six plane angles. 



The problem of the bending of such a polyhedron is a question of trigonometry, and 



58—2 



