93 APPENDIX. . 



Arc?. Lat. Middle point. Degree in Fathoms. 



Punnae and Dodagoontali, ....... ...10 34 49 ...,60496 



Punnae and Bomasimdrum, ... 11 4 44...... 60462 



tfurMae an4 PaugTiwr, -.....*... 11 8 S .......60469 



And, as the two latter arcs give the degree nearly the same, and the 

 latitudes of their middle points differing but little, we may take the mean 

 of the two degrees, as due to the mean latitude of the two middle points s 

 and this will give 50465.5 fathoms due to the latitude of 11 6 23.5. 



2. In order to obtain a mean result between the observations made at 

 Dodagooniah, and those at the stations at Panghur, and Bomasundrum t 

 it will be necessary to determine the ratio of the earth's diameters by 

 using the degree deduced from one or other of these arcs ; and some 

 other degree deduced from recent measurements in northern latitudes. 

 I shall therefore take the degree as given by the observations at Punnae 

 and Dodagoontah, whose middle point is m 10 34 49, and use it with 

 determined by Col. Mudge, for latitude 52 2 20, in order to which 

 be necessary to obtain formulas expressing the ratio of the diame- 

 ters in terms of these degrees and the latitudes to which they apply. 

 That this may be general, let m and m denote the meridional degrees 

 In latitudes / and 7, and let a and b express the equatorial and polar 

 diameters. Then from conic sections and the nature of curvature, it is 

 known that — "LJl - '. -■ is the radius of curvature of the ellip- 



2 \/ Cos. 2 % a* 4 Sin. 2 'I. b z 1 J * 



tic meridian in latitude 7, and — _ % = - _^ the radius of cur- 



' 2 v' Cos.2 ;. a z _j_ Sin.* /. ft.O 3 



vature in latitude /, on that or any other meridian on the same ellipsoid. 

 And since the degrees are as their radii of curvature, we shall have, 



m i m '.'. — .,..'.., -^— , ,- : — - = : = — -^ * - tnat is 



2 V Cos.* 'I. a* 4- Sin. 2 'U.M 3 2 y/ Cos.* I. a* + Sin.* I. b.* J 3 



m' i m :: /Cos. Z L o~~+ Sin. 2 L b z ~\ 3 : /Cos. 2 '/. a z + Sin. 7 % 6*1 3 Of 



