of the Figure of the Earth. 415 



tion of the mass will produce an increase in gravity in passing 

 from the equator to the poles bearing the following ratio to the 

 increase actually observed, 



, 1X 0-00078 3 , lS 



and the number of vibrations lost at Spitzbergen by a seconds' 

 pendulum taken there from the equator, arising from this cause, 

 would be 32(a— 1). 



By the formula already deduced, 



1 _ /3=(a _ 1) |i E |, =(a _ 1) |g = | (a _ 1)neai , y . 



Hence, for any change made in the density of the upper shell, a 

 change half as great again must be made in the lower one to 

 preserve the whole mass constant. Also the ratio of the mean 

 densities of the shells when altered will be 



mass of upper shell volume of lower shell 

 mass of lower shell ' " volume of upper shell 



_ afEj-Eg) 27-8 _3a , 



-£(E 2 -E 3 ) X 64^27 ""4/3 nearly * 



Also, from above, 



5 2. 



a= 3~3 A 



Suppose now that the mean densities of the two shells are made 



3 10 5 



the same ; then /3= -a, and a= -q-, /3= ^, or the upper shell is 



increased in density ith, and the lower diminished ^th ; and 

 the number of seconds lost on this account, by removing the 

 seconds' pendulum from the equator to Spitzbergen =3 J, a 

 quantity perhaps hardly to be detected with certainty. 



If the mean density of the upper shell is made twice as great 



3 4 1 



as that of the lower, then /3= ~ a, a = -, /3= ^ ; or the uppei 



density is increased Jrd, and the lower diminished by one-half ; 



and the seconds lost at Spitzbergen would be 11, which might 



readily be detected. 



Suppose that the density of the lower shell is diminished 



indefinitely so as to make it practically a vast cavern, and its 



mass to be gathered into the upper shell. Then /3 is extremely 



5 

 small, a= q, and the seconds lost at Spitzbergen would be 21. 



10. It will thus be seen from this second test, that the change 

 in distribution of matter relatively to the centre would not have so 



