APPENDIX. 95 



mathematicians, gave the length of the degree 60783 fathoms nearly, 

 for the latitude 46 11 57 ; and if this be substituted for m in the above 

 formula (2) the ratio of the polar to the equatorial diameters will be 

 that of 1 to 1.003370, and therefore the ellipticity -^ nearly. 



The length of the degree at the polar circle in latitude 66 20 12 as 

 determined by the members of the Swedish academy in 1802 and 3, was 

 found to be 60955 fathoms ; and by substituting this for m and retaining 

 the rest of the data, we shall have the ellipticity ^ i ^ nearly. Hence, 

 by reducing these three, the mean ellipticity will be g~ nearly, or the 

 polar to the equatorial diameter as 1 : 1.0032423, the mean result of all 

 the recent measurements. 



5. In order to determine the actual values of a and 6, let m denote 

 the meridional degree in latitude /, as before where the radius of eurva- 



tureis 2vcS^rS ^fW\9 and if A denote the arc l^° &c °) e( l ual 

 radius, we shall have m A = — ■■ - — --^ - . from which arises a 2 b* 



= 2 A m ,/Cos. 2 L a 2 -fSin. 2 n~*~) 3 ; and dividing by a* we get ~ j== ^~ 

 ( Cos. 4 /.+Sin.* L ~ Jl; that is JL~ 2 = h£B ( Cos? I + Sin. 8 k ~y 2 )* 

 which being reduced gives \ a =z m i v . Cn ' t H|f +Si "" T1 3 ? the semiequa- 

 torial diameter. Hence if m = 60487, /=n 6 24 and t+e equal 

 1.0032423, and these substituted in the last formula, we shall have ■§■#== 

 3486906 fathoms ; and as 1.0032423 : 1 :: \ a : -fJii^ ==3475638 fathoms, 

 equal i b. And since -§• a is the radius of the equatorial circle, then 

 \± == 348690 j = 60858 fathoms, the measure of the degree of longitude 

 at the equator. 



6. Since w : m :: v 'cos.»j. + = i = -..sin.* t I : /c<».« $.+===- j-Ws.^% I 

 (3) ; and when m is at the equator, and therefore Sin.* /. .=== and Cos.* 



