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 ellipse 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- 
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 
0+(2)) 
dy 
take the general formula y= , for the radius of cur- 
dx? 
vature of any curve, (See Diff. Cal.), and substitute in it the value 
y2 d? 
of ( ) and of —* drawn from equation (1). By the rules of 
: sph : d 2(1—«a)? d? 
differentiation, equation (1) gives (2) - an sm ah , and Pai 
(l—a)4q? 
~~ 3? which values, together with the equation (2) 
[(1 =2)#(@ ~22)]} 
and the trigonometrical relation between the tangent and sine, will 
a(1—a)? 
(1—sin.2L[1—(1 ~u)2))2 
vature of a terrestrial meridian at any point whose latitude 1, 1s 
give us y= (3), for the radius of cur- 
6. By making in formula (3) L=0°, we shall obtain a(1—“)? 
for the radius of curvature of the meridian where it crosses the 
