Of the Figure of the Earth. 117 



tilde at A=S, the sine of the angle BFl, or of the lati- 

 tude at B=5', radius bein,^ supposed Unity, we shall have 



^'q^/q^—^'^^ +f-^^^'=V7 >^ F=CG^ ; and./- 

 vV'— CI + ^ CI^ =^ V^^"^— Cl2; whence CG^ = 



Z p2 /i2 



-S' f/"^ q'' — s^ q 



s^ /J-, and CI^ = 1 — s'^ + .2. y^z ; also the radius 



qp^ 



of curyature will become R = 1 — s^ -f s^ ^ ^ ; and r — 



— — i 



q P"- 

 i-_5'2-f 52^2\3. From which we obtain this equation R 



X 1— s^ + s^ p^b = r X 1—5^ + s-^ />^ \3- ; from which it is 



evident, that the radius of curvature, or the measure of 

 a degree, which is always in a given ratio to it, is.recip- 

 rocally proportional to the quantity 1 — s^ + s^/j-|2- ; now 



when ^ IS less than Unity, or the equatorial exceeds 



the polar diameter, the terms — s^' + — 4r > ^■^^ negative, 



and the whole expression diminishes in value, as s, or 

 the sine of the latitude increases ; that is, when the equa- 

 torial exceeds the polar diameter, a degixe of the me- 

 ridian increases, in a ratio depending on the sine of the 

 latitude, as determined in the foregoing expression, and 

 vice versa. 



When p=qi or the equatorial and polar diameters are 

 equal, then R=r, or the radius of curvature, or the 

 degree of the meridian, is every where the same. 



As to the actual proportion of these diameters, ac- 

 cording to the measures of a degree, in any tvio latitudes, 



