TRANSFORMATION OF SURFACES BY BENDING. 81 



ness may be neglected. By excluding the thickness altogether, we arrive at 

 Euclid's first definition, which we may state thus 



"A surface is a lamina of which the thickness is diminished so as to become 

 evanescent." 



We are thus enabled to consider a surface by itself, without reference to 

 the portion of space of which it is a boundary. By drawing figures on the 

 surface, and investigating their properties, we might construct a system of 

 theorems, which would be true independently of the position of the surface in 

 space, and which might remain the same even when the form of the solid of 

 which it is the boundary is changed. 



When the properties of a surface with respect to space are changed, while 

 the relations of lines and figures in the surface itself are unaltered, the surface 

 may be said to preserve its identity, so that we may consider it, after the 

 change has taken place, as the same surface. 



When a thin material lamina is made to assume a new form it is said 

 to be bent. In certain cases this process of bending is called development, and 

 when one surface is bent so as to coincide with another it is said to be 

 applied to it. 



By considering the lamina as deprived of rigidity, elasticity, and other 

 mechanical properties, and neglecting the thickness, we arrive at a mathemati- 

 cal definition of this kind of transformation. 



" The operation of bending is a continuous change of the form of a surface, 

 without extension or contraction of any part of it." 



The following investigations were undertaken with the hope of obtaining 

 more definite conceptions of the nature of such transformations by the aid of 

 those geometrical methods which appear most suitable to each particular case. 

 The order of arrangement is that in which the different parts of the subject 

 presented themselves at first for examination, and the methods employed form 

 parts of the original plan, but much assistance in other matters has been 

 derived from the works of Gauss *, Liouvillef, BertrandJ, Puiseux, &c., references 

 to which will be given in the course of the investigation. 



* Disquititiones generates circa superficies curvas. Presented to the Royal Society of Gottingen, 

 8th October, 1827. Commentationes Recentiores, Tom. vi. 



f Liouville's Journal, xn. J Ibid. xnr. Ibid. 



VOL. I. 



