GEODESIC INVESTIGATIONS. 179 



NOTES. 



©The problems solved in this paper are almost all new. Two only (as 

 far as my knowledge extends) have been solved elsewhere or applied 

 in the actual practice of trigonometrical surveys, viz. : — • 



1°. When the given data consists of the latitude of one station S', the 

 azimuth A', from it to the other station S", and the length and circular 

 measure of the geodesic arc between the stations. 



2°. When the latitudes of the two stations are given, and also the difference 

 of longitude of the stations. 



It seems to be the general practice to find methods of determining the un- 

 known latitude, or azimuths, or diiference of longitude only, without paying 

 much respect to the various other entities and the relations subsisting amongst 

 them, by means of which errors of observation can be move easily detected 

 and rectified. 



I have given new methods of solving the two above-mentioned cases. The 

 other equally important cases, which have been hitherto overlooked, will no 

 doubt attract more attention in the future. The whole set, when fully 

 worked out by the various processes suggested by the figure, constitute 

 the basis of a system of Geodesy in which the Equatorial and Polar Radii 

 of the earth are the only standards required, in order to obtain the accurate 

 lengths and bearings of sides of triangles on its spheroidal surface, and the 

 positions of the stations or vertices. 



©In the preliminary portion of the paper many useful theorems and 

 formulae have been evolved. 

 I would direct particular attention to the direct evolution of Dalby's cele- 

 brated theorem ; to the remarkable relation connecting the circular measure 

 2 of the geodesic arc joining two stations, the angle A between the two 

 normal-chordal planes, and the angle f between the two normals to the sur- 

 face at the stations, viz. : — that their halves can be represented by the sides 

 of a right-angled spherical triangle. 



®The errors which have been demonstrated to exist in the investigations 

 and formulse of the " Account of the Principal Triangulation of Grreat 

 Britain and Ireland" relate to the most important questions in practical 

 Geodesy. They pertain to formulse specially recommended as the most exact 

 and reliable, and such as should be employed to test the approximative accuracy 

 of all other methods or formulse. It must be manifest, from what has been 

 shown in the present paper, that they are not fitted for the purjjoses intended, 

 and that they lead to results more erroneous than those arrived at by consider- 

 ing the earth to be a perfect sphere. Some persons may think it strange that 

 those errors were not detected long since in the actual work of the English 

 and Indian Trigonometrical Surveys ; but those who have paid special atten- 

 tion to the science of Geodesy will no doubt at once perceive that it was almost 

 impossible to detect errors in that manner, as all imperfections are generally 

 attributed to " the deviation of the plumb-line from the vertical." 



That the plumb-line must deviate from the normal or vertical line at some 

 stations is evidSnt ; and calculations based on the laws of attraction and 

 measurements of mountain masses can be made, giving a rough estimate of 

 the amount of such deviation ; but, in order to get a correct estimate of its 

 amount and direction, it is, in my opinion, absolutely necessary to take siich 

 observations at each of the principal stations of a triangulation as will enable 

 one to apply all the problems in the pi'esent paper, to every ttvo of the stations, 

 and thus obtain a complete set of entities to which the principles of probabili- 

 ties can be properly applied, instead of confining the calculated results, as is 

 usually done, to what may be given by the application of the first and fifth 

 problems only. 



