460 Mr MAXWELL, ON THE 



Similarly PR may be expressed by 



ds 8 , 



These values of PQ and PR will be the ultimate values of the length and breadth of a 

 quadrilateral facet. 



The angle between these lines will be ultimately equal to (p, the angle of intersection of the 

 system ; but when the values of $u and <$u are considered as finite though small, the angles 

 a, b, c, d of the facets which form a solid angle will depend on the tangential curvature of the 

 two systems of lines. 



Let r be the tangential curvature of a curve of the first system at the given point measured 

 in the direction in which u increases, and let r, that of the second system, be measured in the 

 direction in which u increases. 



Then we shall have for the values of the four plane angles which meet at P, 



1 ds' , I ds „ 



a = IT - <p + r-T cu 7 — cu, 



T 2r du 2r du 



1 ds' , 1 (i(. 

 o = (b + — -— - 6u + — , — cu, 

 r 2r du 2r du 



1 ds' . , 1 ds , 

 C = 7T - d> —> cu + — 7 -— cu, 



T 2r du 2r du 



1 ds , , 1 ds . 



d = v , cu : — cu. 



T 2r du 2r du 



These values are correct as far as the first order of small quantities. Those corrections 

 which depend on the curvature of the surface are of the second order. 



Let p be the normal curvature of a curve of the first system, and p that of a curve of the 

 second, then the inclination I of the plane facets a and 6, separated by a curve of the second 

 system, will be 



1 ds , 

 I = — j— — du , 

 p sin <p du 



as far as the first order of small angles, and the inclination If of b and c will be 



v 1 ds 1 



I = , . — cu 



p sin (p du 

 to the same order of exactness. 



12. On the corresponding polygon on the surface of the sphere of reference. 



By the method described in Art. (2) we may find a point 

 on the sphere corresponding to each facet of the polyhedron. 



In the annexed figure, let a, b, c, d be the points on the 

 sphere corresponding to the four facets which meet at the solid 

 angle P. Then the area of the spherical quadrilateral a, b, c, d 

 will be the measure of the entire curvature of the solid angle P. 



This area is measured by the defect of the sum of the 

 exterior angles from four right angles ; but these exterior angles are equal to the four angles 

 a, b, c, d, which form the solid angle P, therefore the entire curvature is measured by 



k = 27r - (a + b + c + d). 



