TRANSFORMATION OF SURFACES BY BENDING. 103 



and the expression for p becomes 



or if the area of the circumscribing parallelogram be called A, 



I6h 3 



P--JT- 



The principal radii of curvature of the surface are parallel to the axes of 

 the conic of contact. Let R and R denote these radii, then 



= *! and % = \\; 



and therefore substituting in the expression for p, 



1 



or the specific curvature is the reciprocal of the product of , the principal radii 

 of curvature. 



This remarkable expression was introduced by Gauss in the memoir referred 

 to in a former part of this paper. His method of investigation, though not 

 so elementary, is more direct than that here given, and will shew how this 

 result can be obtained without reference to the geometrical methods necessary 

 to a more extended inquiry into the modes of bending. 



14. On the variation of normal curvature of the lines of bending as we pass 

 from one point of the surface to another. 



We have determined the relation between the normal curvatures of the 

 lines of bending of the two systems at their points of intersection ; we have 

 now to find the variation of normal curvature when we pass from one line of 

 the first system to another, along a line of the second. 



In analytical language we have to find the value of 



- 



du 



Referring to the figure in Art. (11), we shall see that this may be done 

 if we can determine the difference between the angle of inclination of the 

 facets a and b, and that of c and d : for the angle I between a and b is 



7 1 ds' > 

 I = . . -V-, Su , 

 psunp au 



