96 The Astronomer Royal on a Problem of Geodesy. 



platinum, 40 for platinized platinum, whilst I found it = 24; 

 with the gas battery. 



I should wish on this point to recall the attention of the 

 reader to the remarks of this philosopher* upon the predo- 

 minating action of secondary batteries over that of the gas 

 battery. 



X. Tlie Astronomer Royal on a Problem of Geodesy. 

 To the Editors of the Philosophical Magazine and Journal. 



Gentlemen, 

 A KNOWLEDGE of the result of the following investi- 

 "^*- gation is so necessary for the correct calculation of the 

 large triangles of modern surveys, and the investigation itself 

 is so easy, that I imagine that an equivalent solution must have 

 appeared already. 1 have not however seen one, and I there- 

 fore think it possible that the publication of my investigation 

 may not be entirely without utility. 



The problem which I propose to myself is " To find the 

 curvature-excess upon a surface differently curved in different 

 directions;" the term curvature-excess having the same mean- 

 ing for this surface which spherical-excess has for a spherical 

 surface ; namely, the quantity by which the sum of the three 

 angles of a triangle (as observed with a theodolite whose axis 

 at each angle of the triangle is made perpendicular to the 

 surface) exceeds 180°. And before entering upon the inves- 

 tigation, 1 wish to point out that the " geodetic lines " and 

 *' geodetic triangles " of speculative geometers have nothing 

 to do with this problem. The triangles with which we are 

 concerned here are the rectilinear triangles formed by the 

 rays of light, proceeding in straight lines (except so far as they 

 are influenced by refraction, which does not sensibly affect 

 their azimuths) from one station to another. The best geo- 

 meters possessing a practical acquaintance with surveys, — 

 Dalby, Delambre, and Everest, — have fully understood the 

 difference. 



I must also point out, that, unless we proceed to an exces- 

 sive degree of complication, it is impossible to give the solu- 

 tion with the same degree of completeness as for a spherical 

 triangle ; because a surface is not defined by a mere knowledge 

 of the principal radii of curvature for one point of the surface, 

 for a higher order of superficial distances from that point than 

 the second. Thus, to take a very simple case, a surface having 

 given principal radii of curvature may be either at the equa- 

 * Loc. cit. vol. Ixi, p. 600. 



