340 



D. Let its intersection with the sphere be the curve c, and 

 let the supplemental cone and corresponding spherical curve 

 be C and c'. 



Then the total curvature of the proposed portion of the 

 given surface S is equal to the portion of the spherical surface 

 included by the radii drawn parallel to the normals to S along 

 the curve B. But as the plane of two consecutive sides of 

 the cone C is parallel to the plane of two consecutive genera- 

 trices of the developable D, and as this latter plane touches 

 the surface S at a point on the bounding curve B, it follows 

 that the area of the curve c represents the total curvature of 

 the proposed part of the surface S. But the area of c is 

 equal to 2 ir, diminished by the perimeter of the curve c. And 

 as the angle between two consecutive sides of the cone C is 

 equal to that between two consecutive generatrices of D, 

 which remains unaltered by development, it follows that the 

 total curvature of the proposed portion of S is equal to four 

 right angles diminished by the angle through which g must 

 turn as it assumes all the positions of tangency to the deve- 

 loped edge of regression of D, until finally it comes into the 

 position of g. And thus the theorem is proved. 



The right lines g and g' being equally inclined to the initial 

 and final elements of the developed curve, the angle between 

 the tangents to these elements is equal to that between g and 

 g ' . We may therefore represent the total curvature of the pro- 

 posed portion of the surface S by the angle between the tan- 

 gents to the initial and final elements of the developed curve ; 

 or, what is the same thing, by four right angles diminished by 

 the entire angle through which the tangent to the developed 

 curve is turned as it passes from its first to its last position. 



Many interesting corollaries may be deduced from the pre- 

 ceding theorem. 



If the proposed boundary B be a closed geodetic curve re- 

 turning into itself, the total curvature of the included portion 

 of the surface will be equal to a hemisphere. And hence, if 



