AND THE FIGURE OF A HOMOGENEOUS PLANET. 
129 
~ S’ ”(S + ^)’ ~(S+/*) : 
and the condition that the resultant of these forces is perpendicular to the 
surface of the fluid is expressed by this differential equation, 
d Tx dx +^ d y + lk dz +f(y d y + zdz ) = 0; 
and the integral, viz. 
c = v + £(/ + * 2 ), 
is the equation of the surface of the fluid in equilibrium. This is incontestably 
the true equation of the surface in equilibrium, since all the forces in action at 
that surface have been taken into account. 
Using x, y, z to represent generally the co-ordinates of any particle of the 
mass, and the symbol V, to denote the function of x, y , z, which is equal to 
the sum of the quotients of all the molecules of the mass of fluid divided by 
their respective distances from the particle, it will be convenient to have some 
means of pointing out whether V belongs to a point in the surface, or to one 
differently situated. For this purpose we shall put r = ^/x 2 + y 1 + z 1 for the 
distance from the centre of gravity, and shall write V (r) for the value of V 
relatively to a point within the mass ; and we shall suppose that r becomes R 
at the upper surface, so that V (R) will denote the value of V for a point in that 
surface. According to this notation, the foregoing equation of the surface of 
the fluid in equilibrium, will be thus written, 
c=v(R ) + 4(y 2 + * 2 )- (i) 
The attraction of the whole mass and the centrifugal force, which are the 
only forces that urge a particle in the upper surface, likewise act upon every 
particle in the interior parts of the fluid. It will contribute to perspicuity if 
to these forces we give the name of the principal forces , in order to distinguish 
them from any other forces which an attentive examination may enable us to 
detect. Assuming any molecule in the interior parts, r being its distance from 
the centre of gravity, and x, y, z its coordinates, we have only to proceed as 
before, writing V (r) for V, in order to find the resolved parts of the principal 
MDCCCXXXI. 
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