55o 
Mr. Airy on the 
density : then finding that sum, and applying to it the con- 
ditions of equilibrium, the values of the constants will be 
found. As we propose to carry our approximation to the 
second order of the difference from a sphere, or the second 
order of the ratio of the centrifugal force to the force of gra- 
vity, it is evident that, without something to guide us, this 
will be a work of considerable labour. 
(2. ) Here, however, we shall derive some assistance from 
former investigations. Clairaut and Laplace have shown 
that, to the first order, the form of every surface of equal 
density is an elliptic spheroid : the difference, consequently, 
of any surface of equal density from an elliptic spheroid is 
only of the second order. If a be the polar semi-axis of an 
elliptic spheroid, a (1 -f- e) the equatorial semi-axis, p the 
sine of the latitude of any point, (the latitude being that which 
is usually termed the corrected latitude), R the radius drawn 
to that point, then R = a (1 -j- e. 1 — y! 2 — yl 2 — The 
radius then of a surface of equal density is a (1 -f- e . 1 — - 
If-. p /2 — ^ /4 J + a quantity of the second order. Now, upon 
using the elliptic value of R, it would be found that the 
equation of equilibrium could not be satisfied, in consequence 
of the appearance of p 1 : but no higher powers of y! would 
enter into that equation. To enable us to take away these 
terms, R must be increased by a function of p, containing 
none but the even powers of p' as far as p*. The most con- 
venient form that we can take is a.A(jt 4 — ^ ,2 ), since it 
vanishes both at the pole and at the equator, and at middle 
latitudes expresses the depression of the surface below the 
ellipsoid whose axes are the same. The value of R then 
