The Astronomer Royal on a Problem of Geodesy. 97 



tor of an oblate spheroid, having the greater curvature per- 

 pendicular to the equator, or at the equator of a prolate sphe- 

 roid, having the greater curvature in the equator: but in the 

 former case, on pursuing the line of greatest curvature, we find 

 the curvature gradually diminishing; whereas in the latter 

 case, on pursuing the line of greatest curvature, we find the 

 curvature continuing invariable. It may also be a portion of 

 an infinity of other surfaces, the nature of which, on departing 

 to a sensible distance from the original point, varies in an in- 

 finity of different ways. 



It is necessary, therefore, to confine ourselves to an ap- 

 proximate solution; and it will suffice to use the approximate 

 formula3 (including with the utmost exactness the terms de- 

 pending on the second power of the sides of the triangle, but 

 no more) which are applicable to the reduction of observed 

 angles. 



Now any surface-triangle, however placed, may be divided 

 into four right-angled surface-triangles (of which one is nega- 

 tive), the sides embracing the right angle being in the direc- 

 tion of the greatest and least curvatures respectively. It will 

 be sufficient to investigate the curvature-excess for one of 

 these right-angled triangles. For it will be found, as a result 

 of the investigation, that the curvature-excess which we are 

 seeking will be expressed by a simple multiple of the area of 

 the right-angled triangle : and as the curvature-excess of the 

 large triangle is readily seen to be the algebraical sum of the 

 curvature-excesses of the four right-angled triangles, and as 

 the area of the large triangle is the algebraical sum of the 

 areas of the four right-angled triangles, it follows that the cur- 

 vature-excess of the large triangle will be the same multiple 

 of the area of the large triangle. 



The process which I shall use is, to compute the reduction 

 of each of the horizontal angles to the chord-angle; the sum 

 of these reductions will be the curvature-excess. The well- 

 known formula for reduction is this: if D and D' be the an- 

 gular depressions of two signals (expressed in parts of the 

 radius), E the horizontal angle included between them ; then 

 the reduction (expressed in parts of the radius) is 



1)2-1- n'2 



D . D'. cosec E - ^— ti^ cot E. 



Let v, w, X be the three sides of the right-angled triangle, 

 XI and w including the right angle ; and let V and W be the 

 radii of curvature in the directions of v and la. Let the angles 

 opposite to i;, w, x be called A, B, C (C being =90"). In 



Phil. Mag, S. S. Vol. 86. No. 24- L Feb. 1 850. H 



