EAR 



EARTH, figure of. ( Play fair > Maddy.) 



1. To find the radius of curvature at any point of the terrestrial meri- 

 dian, supposing the earth to be an oblate spheriod. 



Let a and b be the Equatorial and Polar % axes, r the rad. of curv. to 

 the latitude X, c = a b compression, m =. 57o. 2957795 the number 

 of degrees in an arc radius ; then 



r a 2 c + 3 c sin. 2 X. 



c 3 c 

 or = a cos. 2 x. 



and if D = length of a degree in lat. X, r = m D 



as. c 3 c \ 



..D = (I cos. 2 X. ) 



m \ 2a 2a J 



Cor. 1. At the Equator m D = a 2 c ; at the Pole m D = a -f c; and 

 in lat. 45o. = a c. Hence if E, P, and M = the degree at the Equa- 

 tor, Pole, and lat. 45. ; M = | (P + E). 



Cor. 2. The excess of a degree in any lat. above that at the Equator, or 

 D E, varies as sin. 2 X. 



2. The lengths of two degrees of latitude, of which the middle points 

 are in given latitudes, being known by admeasurement, the Equatorial 

 and Polar diameters of the earth may be calculated from the following 1 

 formulae. 



Let D and D' be the given degrees (the least, or that nearest the Equa- 

 tor being D) X and X' the latitudes of their middle points, then 



m. CD' - D.) 



3 sin. (X' + X) X sin. (X' X)' 



and the compression, or ellipticity of the earth 



_ e D' D 



~~ ~a ~~ 3 D. sin. (X' + X) X sin. (X' X)' 



from which two equations a and r, and consequently a and (, mny be 

 found. 



