in the length of the pendulum vibrating seconds . 421 
diminution which depends on the elliptical form of the earth, 
follows very nearly the same law ; therefore the increase of 
gravitation in proceeding from the Equator to the Pole, may 
be taken as the increase of the square of the sine of the lati- 
tude;* and this will also express the corresponding variation 
in the length of the pendulum. 
Let E = The length of the pendulum vibrating seconds at 
the Equator. 
d— The difference between the length at the Equator 
and at the Pole. 
m = The length of the pendulum in the latitude L. 
n = The length of the pendulum in the latitude L'. 
Then from what has been stated, 
m — E -j- d . sin 2 L 
?r=E-f-i. sin 9 Id 
m — n = (E -f- d sin* L) — (E -f- d sin 2 L') = d (sin 5 L— sin 0 L') 
Hence d ■ ■ — ■ , . . t , f < \ 
sin (L-f-L ) X sin (L — L ) 
and E = m — (d sin 2 L.) 1 
Therefore — expresses the diminution of gravity from the 
Pole to the Equator, which being subtracted from of the pro- 
portion of centrifugal force to gravity at the Equator, will give* 
the ellipticity of the spheroid. 
The centrifugal force at the Equator is expressed by the 
deflection of a point on its surface from the tangent, in one 
second of mean solar time. This is equal to the versed sine 
of 15", 041 8, the arc which the earth describes in its diurnal 
revolution in one second ; and taking the radius of the Equator 
at 3967,5 miles, is found to be ,055696 of a foot. 
* The sin 1 -f the cosine 1 is a constant quantity, equal to the radius 1 , consequently 
as the cosine 1 diminishes, the sine 1 must increase, and vice versa. 
