56 OSCILLATIONS OF THE PENDULUM. [SECT. vr. 



centrifugal force is equal to the 289th part of gravity. Now, 

 it is proved by analysis that whatever the constitution of the 

 earth and planets may be, if the intensity of gravitation at the 

 equator be taken equal to unity, the sum of the compression 

 of the ellipsoid, and the whole increase of gravitation from 

 the equator to the pole, is equal to five halves of the ratio of 

 the centrifugal force to gravitation at the equator. This 

 quantity with regard to the earth is f of 559, or T f 3 . 2 . Conse- 

 quently, the compression of the earth is equal to 1T ' 5 .5 diminished 

 by the whole increase of gravitation. So that its form will be 

 known, if the whole increase of gravitation from the equator 

 to the pole can be determined by experiment. This has been 

 accomplished by a method founded upon the following consi- 

 derations : If the earth were a homogeneous sphere without 

 rotation, its attraction on bodies at its surface would be every- 

 where the same. If it be elliptical and of variable density, 

 the force of gravity, theoretically, ought to increase from the 

 equator to the pole, as unity plus a constant quantity multi- 

 plied into the square of the sine of the latitude (K 126). 

 But for a spheroid in rotation the centrifugal force varies, 

 by the laws of mechanics, as the square of the sine of the 

 latitude, from the equator, where it is greatest, to the pole, 

 where it vanishes. And, as it tends to make bodies fly off the 

 surface,it diminishesthe force of gravity by a small quantity. 

 Hence, by gravitation, which is the difference of these two 

 forces, the fall of bodies ought to be accelerated from the 

 equator to the poles proportionably to the square of the sine 

 of the latitude ; and the weight of the same body ought to 

 increase in that ratio. This is directly proved by the oscilla- 

 tions of the pendulum (N. 127), which, in fact, is a falling 

 body ; for, if the fall of bodies be accelerated, the oscillations 

 will be more rapid : in order, therefore, that they may always 

 be performed in the same time, the length of the pendulum 

 must be altered. By numerous and careful experiments, it is 

 proved that a pendulum, which oscillates 86,400 times in a 

 mean day at the equator, will do the same at every point of 



