of the Pendulum made by Capt. Kater, M. Biot, fyc. 163 



Now, by the doctrine of central forces, if <p denote the cen- 

 trifugal force; tr the circumference of a circle to diameter 

 unity ; r the radius of the given circle in which a body re- 

 volves ; t the time of revolution, and g the gravitating force ; 



then $ — — 11 . But by the theory of the pendulum, if / is 



its length, g= n^l; hence by substitution 



The ratio of the centrifugal force to gravity may be ex- 

 pressed by -^j (5) 



The ellipticity or flattening of the earth is from theory 

 equal to f of the ratio of the centrifugal force to gravity, di- 

 minished "by the fraction obtained from dividing the difference 

 of the lengths of the pendulum at the pole and equator by its 

 length at the equator*. Wherefore if s denote the ellipticity, 

 we obtain 



But by substituting the value of <p from equation (4) 

 -IX— --r "f . (7) 



. '+60 ' 



As / in our investigations denotes the time the earth takes 

 to perform a rotation about its axis, and is found to be 23 h 

 56™ 4 s *0908 = 86164 s -0908; consequently \t = 43082 s -0454, 



and (^-Y= 1856062635 nearly, whence 



e —lx - £- (8) 



2 r+ 1856062635 1 z v ; 



Since ;• is the radius of the equator in this case, I the length 

 of the pendulum there, and y the excess of the length of the 

 pendulum at the pole above that at the equator, we must as- 

 certain the values of these quantities before the ellipticity can 

 be obtained. 



Playfair in his Outlines of Natural Philosophy, vol. ii. gives 

 for the radius of the equator about 20921153 feet; and Col. 

 Lambton in the Philosophical Transactions for 1818 gives 

 60848 fathoms, or 365088 feet, for a degree on the equator. 



The radius is therefore = ^ ^ = 2091800feet. A mean 



between this and Playfair's is 20919576 feet, which may be 



* Mccaniquc Celeste, liv. iii. § 34. 



X 2 considered 



