462 Mk MAXWELL, ON THE 



and the expression for p becomes 



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



l6h 2 



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 



„ lo 2 , . 1 b 2 

 R = - - and R' = - - ; 

 2 A 2 A 



and therefore substituting in the expression for p, 



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 show 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 



111) 



du \pl 



Referring to the figure in Art. (ll), 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 



I m — r—r j-> Su , 

 p sin (p du 



and therefore the difference between the angle of a and b and that of c and d is 



U = — Su = — f ; — - — -, ) SuSu ; 



du du \p sin (p du J 



whence the differential of p with respect to u may be found. 



We must therefore find Si, and this is done by means of the quadrilateral on the sphere 

 described in Art. (12). 



