in the length of the pendulum vibrating seconds. 419 
probably increased in density from the surface to the centre, 
the ellipticity being consequently somewhere between -~- 
and 2 3 0 * 
As it appears then that the ellipticity of the earth varies 
with any difference in the diminution of gravitation from the 
Pole to the Equator, and that this last depends in its turn on 
the ellipticity ; it might have been supposed that any attempt 
to arrive at the figure of the earth in this way must have been 
hopeless. 
But it was reserved for Clairaut to remove this difficulty. 
He found that however the density of the earth be supposed 
to vary, the fraction expressing its ellipticity increases as the 
fraction expressing the diminution of gravity from the pole to 
the equator diminishes, and vice versa ; and in his admirable 
work on the figure of the earth, he has demonstrated this 
beautiful and important theorem; that the sum of the two 
fractions expressing the ellipticity and the diminution of gravity 
from the Pole to the Equator, is always a constant quantity, and 
equal to j of the fraction expressing the ratio of centrifugal force 
to that of gravity at the equator. 
If then the decrease of gravity from the Pole to the Equator 
can be discovered, and it be subtracted from this constant 
quantity, the remainder will be the fraction expressing the 
ellipticity of the spheroid. 
The diminution of gravity may be known by finding the 
difference of the lengths of the two pendulums vibrating in 
equal times at the Pole and at the Equator, as it may be easily 
demonstrated that the lengths of such pendulums are to each 
other directly as gravitation; or, if an invariable pendulum, 
such as I have used, be employed, the squares of the observed 
