149] CLAIRAUT'S THEOREM. 407 



Accordingly the ellipticity, or flattening (aplatissement, Abplattung), 

 denned as the ratio of the difference of axes to the greater, is 



r e~ r p ZK 

 148 ) e= = 



The quantity c = ^ is equal to the ratio of the centrifugal accel- 

 eration G) 2 a at the equator to the acceleration of gravity ^^ at the 

 same place, while n is equal to the ratio of the excess of polar over 

 equatorial gravity to the latter. Thus equation 148) gives us Clairaut's 

 celebrated theorem, 



149) e + n = ~c. 



, Polar gravity equatorial gravity 

 Elhpt^c^ty of Sea -level -\ -- 



Equatorial gravity 



5 Centrifugal acceleration at equator 

 2 Gravity at equator 



The values of the constants in 145) adopted by Helmert as best 

 representing the large number of pendulum observations that had 

 been made up to 1884 are given by 



150) g = 978.00 (1 + 0.005310 sin 2 9), 



agreeing closely with the formula given on p. 33. The value of the 

 centrifugal acceleration is known from the length of the sidereal day, 

 the time of the earth's rotation, giving 



86,164.09 sec. 



and the earth's equatorial radius, given by Bessel as 6,377,397 meters. 

 From this is found 



c = 0.0034672 = ^^ 

 giving by 149) 



e = ~ x 0.0034672 - 0.005310 = 0.0033580 = ^- 



& u<u i .c5U 



By a still closer approximation Helmert finds e = 2Q9 26 - By a 



remarkable chance this coincides almost exactly with the value given 

 by Bessel as the result of measurement of arcs of meridian. A third 

 way of deducing the ellipticity is by means of the precession of the 

 equinoxes, which, as has been shown in 96, enables us to calculate 

 the ratio ~ , 



O A. 1 



C = 297 



from which, though involving an assumption as to the distribution 

 of density in the earth, the ellipticity may be derived. Finally, as 



