302 



Colonel A. R. Clarke on a Correction 



nished energy would prevent my engaging in any scientific 

 controversy, were any called for. 



Robert Mallet. 



Loudon, September 17, 1877. 



XL. On a Correction to Observed Latitudes. 

 By Colonel A. R. Clarke, C.B., F.R.S. 



I do not know tliat any demonstration has ever been given 

 of a theorem enunciated by Gauss in a letter addressed 

 by him in 1853 to General Baeyer. It appears from the letter 

 in question (of which a translation appeared in the Comptes 

 RendiLs des Stances de la Commission 2')ermanente de V Associa- 

 tion Geodesique internationale tenues a Florence en 1869) that, 

 at the suggestion of General Baeyer, Gauss had investigated 

 the influence of altitude above the sea-level on observed lati- 

 tudes, and had arrived at a result which, if not very important 

 to geodesists and astronomers, still is very curious and inter- 

 esting. The point in question is this : — In the adjoining dia- 

 gram let A P E be a qua- 

 drant of a meridional sec- ^H' 

 tionofthe earth, NPP^ 

 the normal at any point P. 

 Let P N represent gravity 

 at P ; and on the same 

 scale let P H, parallel to 

 the equator C E, represent 

 the centrifugal force at 

 the same point; then com- 

 pleting the parallelogram 

 GH, PG will be the at- 

 traction of the earth. At 

 a point P^ vertically above 

 P the centrifugal force 

 P His increased to P^H^, 

 retaining its parallelism ; 

 and the attraction P G is 



diminished to P^ G^ — not, however, parallel to its former di- 

 rection, for GP G^P^ converge towards the zenith. The 

 resultant gravity P^ W at P^ is therefore no longer of neces- 

 sity in the line P^ P : and the point to be determined is the 

 magnitude of the angle P P^ W. 

 The potential of the spheroid (.i-' + ?/-)(l-2e) + c- = c2, whose 



* Communicated bv the Author. 



