FIGURE OF THE EARTH. 99 



ply this number by QO' 1 , we get one-fourth part of the meridian Formulae em- 

 Vi , ployed in the 



of the earth. preceding 



The correction to be deduce! for oblateness is 58, 59, or computation. 

 61 toises, according as it re assumed to be j-V~, — V-» or -y-j-o-, 

 and if we take the mean of these, we have the fourth part of 

 the meridian Q= 5130886 tdises ; and hence the metre =a 

 4433080/ lines j so that the value of the metre turns out to 

 be almost entirely independent of the ellipticil form of the 

 earth. 



The radius of the equator is derived from the expression 



log. a = log. ( 2 ^-) + K . (I . « f T V . * 2 - ^.s 3 ), e being 



the oblateness, and ?r the periphery of a circle = 3, 141 6. 



In order to compare any degrees measured with those ob-e 

 tained on the elliptic hypothesis, we have, a very simple for-* 

 mula. Let m and m. be the values of two degrees on .the. 

 meridian, of which the mean latitudes are .4,1 and -4-2 ; in com- 

 paring the analytic expressions for these two degrees deve- 

 loping them, and then making ^ = 45°, we have rri = m . 

 (1— i . p . cos. 2^2+g . cos. 2 2l2), m a? 5/010,5 toises, p — 



i * r, { , H v . ^j° . and > g w%r- <?■ ry^t-, ) 



•* v ■* J l' J . sin. 1 ° c * ^1 .sin. 1"/ 



And then we shall find that the oblatene3s yfa gives 57075,66 

 and 57192,38 toises for the degrees in England and Lapland. 



I shall here subjoin one reflection more, which appears of 

 importance. The oblateness of the earth is a quantity which 

 varies considerably, by the least difference in the elements on 

 which it depends. Accordingly, it is not surprising, that its 

 value fluctuates between two proportions which differ sensibly 

 from each other. To illustrate this, let p be the function 

 which serves to determine the oblateness of the earth, so that 

 7- = p. When this equation varies — h = e 2 . $p. 



Now the coefficient g 2 being very great, we see why the 

 least variation in the elements of the function p, occasions so 

 considerable a variation in the denominator of the oblateness. 

 This is precisely what happens in the lunar equations depen- 

 dent on the figure of the earth, and which M. Laplace has 

 deduced from his beautiful theory. Thus, for example, in the 

 inequality that depends- on the longitude of the moon's node, 

 which he has determined analytically with so much precision, 

 ft 2 the 



