102 



TRANSFORMATION OF SURFACES BY BEXDIKO. 



By the expression for I and T in the last article, we find 



1 ds ds' s s , 



K = j -j- -j-, OU OU 



pp sin 9 (fit au 

 for the entire curvature of one solid angle. 



Since the whole number of solid angles is equal to the whole number of 

 facets, we may suppose a quarter of each of the facets of which it is composed 

 to be assigned to each solid angle. The area of these will be the same as that 



of one whole facet, namely, 



. ,dsds'~ ~ , 

 sin d> -j- -j, ou ou ; 

 ^ du du 



therefore dividing the expression for k by this quantity, we find for the value 

 of the specific curvature at P 



which gives the specific curvature in terms of the normal curvatures of the 

 lines of bending and their angle of intersection. 



13. Further reduction of this expression by means of the " Conic of Con- 

 tact," as defined in Art. (3). 



Let a and b be the semiaxes of the conic of contact, and h the sagitta 

 or perpendicular to its plane from the centre to the surface. 



Let CP, CQ be semidiameters parallel to the 

 lines of bending of the first and second systems, and 

 therefore conjugate to each other. 



By (Art. 3), 



and p = 



and the expression for p in Art. (12), becomes 



But CP . CQ sin < is the area of the parallelogram CPRQ, which is one 

 quarter of the circumscribed parallelogram, and therefore by a well-known 

 theorem 



