Mr. Ivory on the Ellipticity of the Earth. 93 



rate directly and limit the accuracy of the coefficients sought, 

 the same quantum of error, however, having more or less in- 

 fluence according to the latitude of the experiment. In this 

 latter research, therefore, it is requisite to have a number of 

 independent experiments, and to combine them, by the known 

 analytical methods, so as to give to each error its due weight. 

 By this process we deduce the most advantageous formula for 

 computing the length of the pendulum in any latitude, which, 

 again, necessarily involves in it a mean determination of the 

 ellipticity. Thus we may employ two different ways for finding 

 the ellipticity of the earth from experiments with the pendu- 

 lum ; and consistency requires that both methods should agree 

 in leading to the same result, more especially if in both cases 

 the same data have been used. Our confidence in the good- 

 ness of the methods, and in the accuracy of the results, must, 

 it is evident, be strengthened or weakened, according as they 

 concur or fail to do so. 



In what follows I shall endeavour to determine the ellipti- 

 city of the earth, by each of the two methods above mentioned 

 separately, using the best sets of experiments with the pen- 

 dulum that have hitherto been made. 



1. Let e denote the earth's ellipticity; <p = — ^— , the pro- 

 portion of the centrifugal force to gravity at the equator ; G 

 the gravity at the equator ; and g the like force at the latitude 

 A: then g = G \l + ( 5 -f- e) sin* K ^ 



or, esin a A= 1 + 4rsin 2 X~ 4-. 



Again, if L be the equatorial pendulum, and I the pendu- 

 lum at the latitude A ; then, because the length of the pendu- 

 lum is proportional to gravity, we have, 



^sin 3 A= 1 + ^-sin s A- ~; 



and, e = -00865 — *T L . (1) 



7 L sin 2 X \ l J 



If we knew the length of the equatorial pendulum, we should 

 thus obtain a value of e for every experiment that determined 

 I and A. And as the earth's ellipticity, and the total increase 

 of the seconds pendulum from the equator to the pole, are 

 both known within certain limits, we may derive the value 

 of L from a pendulum found by observation very near the 

 equator. Thus at Maranham, in latitude 2° 31' 43" = a', 

 the seconds pendulum, as determined by Captain Sabine, is 

 39-01214- in. = V ; the excess of the polar above the equatorial 

 pendulum can hardly be greater than 0-21 in., or less than 

 0*2 in,; wherefore, 



