224 Length of a Degree of the Terrestrial Meridian. 



can be drawn tangent at the same point of any curve, that which 

 coincides with the curve for the greatest extent, is called the oscu- 

 latory circle of the curve ; and the radius of this circle is called the 

 radius of curvature of the curve. By the radius of curvature we 

 may judge of the degree of curvature of the curve at its different 

 points ; for the curvature at the point of contact being the same as 

 that of the osculatory circle, and the curvature of a circle being 

 greater as its radius is less, and vice versa, it follows, that the cur- 

 vature of a curve is greater as its radius of curvature is less, and 

 vice versa. In the elhpse the curvature is a maximum, and the 

 radius of curvature a minimum, at the extremities of the major-axis : 

 in going from these extremities towards the flattened parts of the 

 curve, the curvature decreases, and the radius of curvature increases, 

 until we arrive at the extremities of the minor axis, where the cur- 

 vature becomes a minimum and the radius of curvature a maximum. 

 So, in going on a meridian towards the poles, the radius of curvature, 

 being least at the equator, increases from one latitude to another, 

 until we arrive at the poles, where the curvature is the least, and 

 the radius of curvature the greatest. 



. 5. For the purpose of expressing the forementioned circumstances 

 attending the curvature of a terrestrial meridian in a formula, we 



take the general formula y= , for the radius of cur- 



vature of any curve, (See DifF. Cal.), and substitute in it the value 



(dii^\ d^y 



■4- I and of -7-^3 drawn from equation (1). By the rules of 



IdyY a;2(l-a)2 d^'y 



differentiation, equation (1) gives \^^j — ^2 _^T~5 and ^ = 



(l-a)4a2 



^j which values, together with the equation (2) 



\{\-aY{a-'-x^)'f 



and the trigonometrical relation between the tangent and sine, will 



give us 7= ^ 55 (3), for the radius of cur- 



(l-sin.24.[l-(l-a)2])^ 



vature of a terrestrial meridian at any point whose latitude ^^ is 

 known. 



6. By making in formula (3) 4^=0°, we shall obtain a(l— a)^ 

 for the radius of curvature of the meridian where it crosses the 



