258 



Arclideacon Pratt on the effect of 



the length determined in paragraph 4 for latitudes, are never used in sur- 

 vey operations : the great arcs are always divided into much smaller por- 

 tions. Hence the maps constructed from geodetic operations will always 

 be relatively correct in themselves ; but the precise position of the map on 

 the terrestrial spheroid will be unknown by the amount of the unknown 

 deflection of the plumb-line in latitude and longitude at the place which 

 fixes the map. In India the ejffect of the Himmalaya Mountains and the 

 Ocean, taken alone, would throw out the map by nearly half a mile. The 

 calculations, however, which I give in the next two sections of this paper, 

 show that the effect of variations in the density of the crust below almost 

 entirely counteracts that of the mountains and ocean at Damargida in lati- 

 tude 18° 3' 15", and the displacement of the map is almost insensible if 

 fixed by that station. If fixed by the observed latitude of any other station, 

 the map will be out of its place by the local deflection of the plumb-line at 

 that station. This, in the Indian Great Arc, does not exceed one-thirteenth 

 of a mile at any of the stations w^here the latitude has been observed. It 

 appears also from those calculations, that, except in places evidently situated 

 in most disadvantageous positions, the local attraction is rarely of any con- 

 siderable amount. 



§ 2. Hffect of Local Attraction on the Determination of the Mean 

 Figure of the Earth. 

 6. The mean radius of the earth is nearly 20890000 feet, the ellipticity 



is nearly——, and it is found convenient to put the semiaxes of the earth's 



figure under the form 

 a-^h __( u ^ 

 ~2~ ~\ ~ 10000, 



20890000=20890000-2089 M feet,' 



-U'-X^o + ^) ^«««««««= ^''''''')-' 



u and V are quantities to be determined, and the squares and product 

 of these may be neglected. 



Also, ellipticity ="-=i = ^(l + ^). 



The arcs which are actually measured in geodesy do not necessarily 

 belong to precisely the same ellipse : in fact those arcs may not precisely 

 belong to any ellipse. Suppose one of these measured arcs is laid along 

 the ellipse of which the axes are given above, and that, small corrections x 

 and being added to the observed latitudes of its extremities, the arc 

 with its corrected latitudes exactly fits this ellipse. Then x — x may be 

 expressed in the form m-{-ccu+(3v, where m, a, and /3 are functions of the 

 measured length, the observed latitudes, and numerical quantities. Let 

 this be done for all the arcs which have been measured and their subdivi- 

 sions. I shall take the eight arcs used in the chapter on the Figure of the 

 Earth in the Volume of the British Ordnance Survey ; viz. the Anglo-Gallic, 



