[ 92 j 



XII. On the Ellipticity of the Earth as deduccdfrom Experiments 

 made with the Pendulum. By J. Ivory, Esq. M.A. F.R.S. 



[Continued from p. 10.] 



r T , HE figure of the earth is inferred from the relative force 

 -■- of gravity at different points on its surface ; and therefore it 

 depends upon the proportion of the pendulums that oscillate 

 in a given time, and not upon their absolute lengths. Now, 

 the proportion of two lines, nearly equal, may be known with 

 some degree of precision notwithstanding the existence of 

 considerable errors in the measurement of the lines themselves. 

 If the errors be both in excess or both in defect, they will 

 hardly alter the proportion of the lines ; and if they be of op- 

 posite kinds, one in excess and the other in defect, although 

 they will combine in changing the proportion sought, yet the 

 variation produced will be inconsiderable, supposing that their 

 magnitude is small in comparison of the whole lengths. When 

 the earth's ellipticity is derived from the proportion of two 

 pendulums found experimentally at different places, the accu- 

 racy of the result is commensurate to the accuracy with which 

 the ratio of the pendulums has been determined. The error 

 in one case will increase or decrease exactly at the same rate 

 as the error in the other case. In independent experiments, 

 executed with equal care, there can be no reason for preferring 

 some and rejecting others ; an equal authority must be assigned 

 to all ; and the same thing, it is evident, will be true of the re- 

 sults obtained when such experiments are combined with one 

 another. This, however, must be understood as conclusive 

 only in good combinations of the experiments, that is, in such 

 combinations where the difference of the two pendulums is 

 large enough to cover and, as it were, absorb the unavoidable 

 errors of observation. In this manner of proceeding, all the 

 results having the same weight, the most advantageous deter- 

 mination of the ellipticity will be found by taking the arith- 

 metical mean, in the expectation that the excesses will com- 

 pensate the defects. 



Again, a minute change in the earth's ellipticity produces 

 a great variation in the length of the pendulum ; and, in con- 

 sequence, a variation of the pendulum affects but slightly the 

 quantity of the ellipticity. Hence the effect of the errors of 

 the experiments is very different in the ellipticity, and in the 

 general expression of the absolute length of the pendulum. 

 The first is affected by the errors only in so far as they alter 

 the proportion of two pendulums at distant latitudes ; but, in 

 the expression of the length of the pendulum, the errors ope- 

 rate 



