AND THE FIGURE OF A HOMOGENEOUS PLANET. 
125 
then r will be the line drawn from the centre to the point (x, y, z) ; and the 
arcs 0 and ^ determine the direction of r, 6 being the angle between r and the 
axis of the coordinates parallel to x, and the angle which the plane containing 
r and the same axis makes with the plane of x, y. By substituting, we get 
f = f*( B 1 |* + B a .|S + B 3 ? + B 4 S, + B 5 5S + B 6 „Q 
-f- r 3 (D, |? + D 2 >i 3 + D 3 + D ! 2 !j + &c.) 
+ &c. 
The symbols |, r h represent three rectangular coordinates of a point in the 
surface of a sphere having unit for its radius ; and, in order to simplify, I shall 
write Q 2 , Q 3 , and generally Q„, for homogeneous functions of g, y, of two, 
three, and n dimensions : then, 
<p = r 2 Q 2 + r 3 Q 3 -f r 4 Q„ + &c. 
For the sake of distinction. Jet R represent a line drawn from the centre of 
gravity to the surface of the fluid ; and r a line drawn from the same centre 
to any interior point at which the pressure is p, the directions in which R and 
r are drawn being determined by the arcs 6 and ■ the equation of the fluid’s 
surface, and the expression of p, will be as follows, 
C = R 2 Q 2 + R3 Q 3 + R4 Q n + &c. 
p — C — (r 2 Q 2 + r 3 Q 3 + r 4 Q„ + &c.) 
By means of these equations a radius, R or r, will be known when the arcs 6 
and ^ which determine its direction are assumed ; and in this manner we may 
find all the points of the outer surface, and of any interior level surface in 
which p has any assigned value less than C. All these surfaces will return into 
themselves and inclose a space : because in whatever direction we proceed 
from the centre of gravity to the surface, the function <p passes through every 
gradation of magnitude between zero and the maximum. 
It is now easy to complete the demonstration of the equilibrium. A stratum 
of the fluid between the outer surface and any interior level surface will evi- 
dently be in equilibrium, if we suppose that the level surface maintains its 
figure, or rather, that there are no forces urging the particles contained within 
that surface : for, the upper part of the stratum cuts the resultant of the forces 
