86 TRANSFORMATION OF SURFACES HY l.KM'IMJ. 



When p. is greater than 1 it may happen that for some values of *, -j- is 



creater than . In this case 

 P- 



dr <!> . 



= /i-y-; is greater than 1 ; 



a result which indicates that the curve becomes impossible for such values of 

 s and p.. 



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 appli- 

 cable 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 trigo- 

 nometry, and equations might be found connecting the angles of the different 

 edges which meet in each solid angle of the polyhedron. It will be shewn that 



