NEWTON S PRINCIPIA. 181 



any latitude. Newton now refers to a number of obser 

 vations on the length of the seconds pendulum in various 

 latitudes, as a means of testing the truth of his theory. It 

 appeared that the length of the seconds pendulum does 

 decrease as we approach the equator in the ratio of the 

 square of the sine of the latitude, and so far observation 

 confirms the theory. But it also appeared that the 

 decrease of gravity was greater than that given by the 

 theoretical expression. Whence Newton concluded that 

 the value of e, as given by theory, was a little too small. 

 Here, however, he was wrong; for if the earth be con 

 sidered as heterogeneous, an exactly opposite conclusion 

 will follow from Clairaut s theorem. 



The planet Jupiter, owing to its great angular velocity, 

 is very protuberant at its equator, and thus the difference 

 of the lengths of its two axes could be determined with 

 tolerable accuracy. This planet, therefore, furnished 

 Newton with a good test of the truth of his theory. Since 

 the centrifugal force varies as the square of the angular 

 velocity, and inversely as the radius; and gravity in 

 different planets varies as the radius and as the density 

 conjointly, hence the ratio of the centrifugal force to 

 gravity varies as the square of the angular velocity 

 directly and the density inversely. The ellipticity by 

 what precedes has the same proportion. On substituting 

 the known numerical values of these quantities, we find for 



the ellipticity of Jupiter -^7. Cassini observed in the year 



1691, that the diameter of Jupiter from east to west is 

 greater by about T jth part than the other diameter. Mr. 

 Pound with his 123 feet telescope and an excellent 

 micrometer, measured the diameters of Jupiter in the 

 year 1719, and found them on four occasions to have the 

 ratios 12 to 11, 13| to 12J, 12f to llf, 14^ to 13J. 



K 3 



