100 Meaxurement of three "Degrees of the Meridian. 



I shall here suijoin one n^flection more which appears 

 of inipcrtarice. Tlie oblaltness ot llic earth is a quantity 

 which varies cc)ii.siclerai)lv, hv the lea>t difference in ihe ele- 

 iiien;s on >vhich ii depends. Arcordinorlv it in not siir- 

 prisniff, that its value fluctuates between two proportions 

 which differ sensiblv from each other. To illustrate this, 

 let p l)e the tunction which serves to determine the oblate- 



ness of the earth, so that — = />• When this equation 



varies — 8e = e-. gp. 



Now the coefficient e* being very great, we see why the 

 least variation in the elements of the iunction p occasions 

 so considerable a variation in the denominator of the oblate- 

 ness. Tins is precisely what hapi)ens in the lunar equa- 

 tions dependent on the fiiiure ot the earth, and which 

 M. Laplace has deduced from his beautiful theory. Thus, 

 for example, in the inequali'v that depends on the longi- 

 tude of the moon's node, which he has determined analy- 

 tically with so much precision, the numerical coefficient 

 found bv Burg gives -^-^ for the oblateness ; but \\ this co- 

 efficient be diminished by o",6G5, then the oblateness be- 

 comes ■^\—. so that a variation even to this small amount 

 in the coefficient augments the denominator of the oblate- 

 ness nearly — „ part. 



The same happens with regard to the pendulum vibrating 

 seconds; for, supposing its length at 45° to have been cor- 

 rectly ascertained by MM. Biot and Mathieu, if we^ wish 

 to knoa the length of a second's pendulum at the equator, 

 corresponding to an oblateness of ■^^■^■, we find it to be 

 439, 18'0 lines. Now this length differs trom that deter- 

 mined bv Bonguer only by 0,029 of a lipe, and M. La- 

 place even thinks that the result of Bouguer should be di- 

 minished by about double (ins quantity. We see from 

 hence how much these little diffeiences, whether produced 

 by errors of observation, or irre<j;ularities in the earth itself, 

 are liable to affect the denominator of the fraction cxpress- 

 iuff the oblateness. 



Fortunatelv, it seems probable, that the utmost latitude 

 of our present uncertainty is between the limits of 330 and 

 310, and the mean of these may be considered ^s a very 

 near approximation to the truth. 



XV. On 



