Length of a Degree of the Terrestrial Meridian. 225 



a 

 equator, and by making -vj^^ 90° we shall obtain :j for the ra- 

 dius of curvature of the meridian at the poles : and as a is less 

 than unity, it follows, that the former radius of curvature is less, and 

 that the latter is greater, than the semi-equatorial diameter ; hence, 

 the curvature is greater at the equator and less at the poles, than at 

 any other point on the meridian — which is in accordance with what 

 is said in 4. 



7. Having found the radius of curvature (3) for any point of the 

 meridian, we can construct the osculatory circle at the same point ; 

 and as this circle will be sensibly confounded with the meridian itself 

 for some extent, the length of a degree of the circle, will be sensi- 

 bly equal to the length of the degree of the meridian, at the point 

 of contact. It is upon this principle, that we shall obtain the length 

 of a degree whose middle point is at any place m, in terms of the 

 measure of the earth's oblateness and equatorial diameter; thus — 



8. Denoting by L, the length of a degree of a terrestrial me- 

 ridian, and by -r, the ratio of a circumference to its diameter, we 

 shall have the proportion L : 2ty.:;l° : 360°; whence, by sub- 

 stituting the value of the radius of curvature (3) we derive L= 



Y^x«(i-«)-^ 



-^5 (4), for the formula by which the 



(l-sin.H[l-(l-«)-])' 



length of a degree, having its middle point at a place whose latitude 



4- is known, may be estimated: for examples, making 4'=0° and 



* If 



^|.=:90° in the formula, we shall obtain f5QXa(l — «)= and r^ X 



a 



jYZ. — \2 respectively, for the lengths of the degrees, one at the 



equator, and the other at the pole ; and as a is less than unity, we 



perceive that the length of the former is greater than the length of 



If 

 the latter: moreover, 7q7^X«5 which expresses the length of a de- 



if 

 gree of the equator, being greater than TQ7;Xa(l — a)^. we infer, 



that the length of the degree of the meridian, where it crosses the 

 equator, is less than the length of a degree of the equator itself — a 

 fact which might have been anticipated, from the circumstance of 

 Vol. XXXI.— No. 2. 29 



