The Astronomer Royal on a Problem of Geodesy. 99 

 The sum of the three reductions, or the curvature-excess, is 

 .T^. sin A. cos A mo 



or 



2VW ' 2V W 



The area of the triangle = -— . Hence the expression for 



the curvature-excess is 



Area of triangle 



vw • 



If we had a triangle of the same area upon a sphere whose 



A. 1*63. 



radius is R, the spherical-excess would be p^ . Hence the 



curvature-excess is the same as the spherical-excess of a tri- 

 angle of the same area on the surface of a sphere whose radius 



R=\/VW. 



The application of this theorem to the triangles of a terres- 

 trial survey is very easy. 



One of the deductions from the general expression is, that 

 the curvature-excess upon a developable surface is =0. For 

 there one of the radii of curvature is infinite. This result is 

 very easily verified in the particular case of the cylinder, 

 whatever be the magnitude of the triangle. For if, in the first 

 place, we consider the "geodetic" line upon a cylinder (which, 

 when inclined to the axis, is a regular helix), it will be ob- 

 vious that its directions at its two extremities make equal 

 angles with the axis of the cylinder; and next, if we consider 

 the positions of the planes of reciprocal vision at the two sta- 

 tions (which are different), and their intersections with the 

 cylindrical surface at the respective points of vision (neither of 

 which intersections coincides with the helix), it will be obvious 

 that the directions of these intersections also make equal 

 angles with the axis of the cylinder ; and therefore they are 

 equally inclined (one on one side, and the other on the other 

 side) to the helix ; and therefore the sum of the visual azi- 

 muthal angles in the visual triangle is equal to the sum of the 

 angles in the helical triangle. But on developing the surface, 

 the helical triangle becomes a rectilinear triangle, or its cur- 

 vature-excess = ; and therefore the curvature-excess of the 

 visual triangle =0. 



Royal Observatory, Greenwich, G. B. AlRY. 



January 10, 1850. ^ 



H2 



