454 Mr MAXWELL, ON THE 



The surface is now covered with a network of curvilinear triangles. The plane triangles 

 which have the same angular points will form a polyhedron fulfilling the required conditions. 

 By increasing the number of the curves in each series, and diminishing their distance, we may 

 make the polyhedron approximate to the surface without limit. At the same time the 

 polygons formed by the edges of the polyhedron will approximate to the three systems of 

 intersecting curves. 



2. To find the measure of the " entire curvature'''' of a solid angle of the polyhedron, 

 and of a finite portion of its surface. 



From the centre of a sphere whose radius is unity draw perpendiculars to the planes of 

 the six sides forming the solid angle. These lines will meet the surface in six points on the 

 same side of the centre, which being joined by arcs of great circles will form a hexagon on the 

 surface of the sphere. 



The area of this hexagon represents the entire curvature of the solid angle. 



It is plain by spherical geometry that the angles of this hexagon are the supplements of 

 the six plane angles which form the solid angle, and that the arcs forming the sides are the 

 supplements of those subtended by the angles of the six edges formed by adjacent sides. 



The area of the hexagon is equal to the excess of the sum of its angles above eight right 

 angles, or to the defect of the sum of the six plane angles from four right angles, which is 

 the same thing. Since these angles are invariable, the bending of the polyhedron cannot alter 

 the measure of curvature of each of its solid angles. 



If perpendiculars be drawn to the sides of the polyhedron which contain other solid 

 angles, additional points on the sphere will be found, and if these be joined by arcs of great 

 circles, a network of hexagons will be formed on the sphere, each of which corresponds to a 

 solid angle of the polyhedron and represents its " entire curvature." 



The entire curvature of any assigned portion of the polyhedron is the sum of the entire 

 curvatures of the solid angles it contains. It is therefore represented by a polygon on the 

 sphere, which is composed of all the hexagons corresponding to its solid angles. 



If a polygon composed of the edges of the polyhedron be taken as the boundary of the 

 assigned portion, the sum of its exterior angles will be the same as the sum of the exterior 

 angles of the polygon on the sphere ; but the area of a spherical polygon is equal to the 

 defect of the sum of its exterior angles from four right angles, and this is the measure of 

 entire curvature. 



Therefore the entire curvature of the portion of the polyhedron enclosed by the polygon 

 is equal to the defect of the sum of its exterior angles from four right angles. 



Since the entire curvature of each solid angle is unaltered by bending, that of a finite 

 portion of the surface must be also invariable. 



3. On the " Conic of Contact,'''' and its use in determining the curvature of normal 

 sections of a surface. 



Suppose the plane of one of the triangular facets of the polyhedron to be produced till it 

 cuts the surface. The form of the curve of intersection will depend on the nature of the 



