NEWTON S PKINCIPIA. 341 



place to place, are irregular, and always very small. The 

 mean of all the values of the above fraction may be de 

 fined to be the ellipticity. We cannot observe directly 

 the value of r. In practice, therefore, we replace this 

 fraction by another which contains g instead of r. If we 

 could make a vast number of observations in all parts of 

 the world, no further correction would be necessary than 

 merely taking the mean of all the observed values of s. 

 But as all our observations are made on land, and are few 

 in number, our errors may all tend in one direction : it is 

 therefore necessary to reduce our results to the surface of 

 the spheroid. 



The third method of determining the ellipticity is 

 founded upon astronomical observations. If the earth 

 were a perfect sphere and homogeneous, the elliptic orbit 

 of the moon would only be affected by the known disturb 

 ances of the other heavenly bodies. But the earth is neither 

 spherical nor homogeneous. Hence arise other inequa 

 lities in the moon s motion, and conversely, when these are 

 observed, they will enable us to discover the ellipticity of 

 the earth. Without making any assumption as to the law 

 of density, the theory of the &quot; Figure of the Earth &quot; en 

 ables us to find its attraction on the moon : substituting 

 these in the equations of the moon s motion, we can deduce 

 two inequalities. The chief of these is the inequality in 

 latitude, and is about S&quot;, hence we get e = -003370. 

 The other is the inequality in longitude, and lies between 

 6&quot;-8 and 7&quot;, the former gives e = -003360, the latter 

 e = -003407. 



It is not difficult to see what will be the general nature 

 of the more important inequality. The consequence of 

 the attraction of the ring of matter surrounding the earth 

 at the equator, will manifestly be to pull the moon nearer 

 the equator. The chief disturbance will therefore be that 

 produced by a small force acting on the moon perpen 

 dicular to the plane of her orbit, and tending towards the 

 earth s equator. The effect of such a force is easily seen. 



z s 



