446 



Mr MAXWELL, ON THE 



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

 perties, and neglecting the thickness, we arrive at a mathematical 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 1 , Liouville 2 , Bertrand 3 , Puiseux 4 , &c. references to 

 which will be given in the course of the investigation. 



On the Bending of Surfaces generated by the motion of a straight line in space. 



If a straight line can be drawn in any surface, we may suppose that part of the surface 

 which is on one side of the straight line to be fixed, while the other part is turned about the 

 straight line as an axis. 



In this way the surface may be bent about any number of generating lines as axes 

 successively, till the form of every part of the surface is altered. 



The mathematical conditions of this kind of bending may be obtained in the following 

 manner. 



Let the equation of the generating line be expressed so that the constants involved in it 

 are functions of one independent variable u, by the variation of which we pass from one 

 position of the line to another. 



If in the equation of the generating line Aa, u=u x , then in the equation of the line Bb we 

 may put u=u ll , and from the equations of these lines we may find by the common methods 

 the equation of the shortest line PQ between Aa and Bb, and its length, which we may 

 call 5£. We may also find the angle between the directions of Aa and Bb, and let this 

 angle be §9. 



In the same way from the equation of Cc, in 

 which u=u 3 , we may deduce the equation of RS, 

 the shortest line between Bb and Cc, its length S£ s , 

 and the angle $0 2 between the directions of Bb 

 and Cc. We may also find the value of QR, the 

 distance between the points at which PQ and RS 

 cut Bb. Let QR = &r, and let the angle between 

 the directions of PQ and RS be $<p. 



I 



1 Disquisitiones generates circa superficies curvas. Pre- 

 sented to the Royal Society of Gottingen, 8th October, 18?7. 

 Commentationes Recentiores, Tom. vi. 



8 iiiouville's Journal, xn. 

 3 Ibid. xiii. 

 * Ibid. 



