204 Mr. Airy on the Figure of a Fluid Mass 
given in detail, is restricted to the case in which the disturbing 
forces are symmetrical about an axis; but the extension of the 
same principle, which enables us to apply a similar process to 
any forces whatever, is shortly indicated at the termination. To 
shew the method of applying the theory to any given case, I have 
considered the effect which the ring of Saturn produces on his 
figure (supposing him homogeneous) and arrive at the important 
conclusion, that the attraction of the ring, on the hypothesis of 
gravitation, will not explain the peculiarity which has been ob- 
served in the figure of Saturn. It is not easy to assign any 
other cause for the singularity of form of that remarkable planet ; 
and if the observations remain undisputed, the physical expla- 
nation of this phenomenon may exercise the ingenuity of future 
philosophers. 
(1.) Let a, 6, c, be the rectangular co-ordinates of any point 
of the mass parallel to the axes of x, y, z,: let X, ¥, Z, be the 
forces acting upon that point in the same directions. Then if 
a function U can be found such that 
dU Se 
duieitedda wat hades Te 
the equation to the surface of equal pressure passing threugh 
this point, will be U=C. The condition necessary to the existence 
of equilibrium is, that a function U can be found which will 
satisfy these equations. In the case which we propose to con- 
sider, the forces arise entirely from attractions and centrifugal 
force; and will therefore, by a well-known theorem, satisfy these 
equations. The equilibrium being possible, we have only to find 
the equation of the surface bounding the mass; which we shall 
attempt by the assistance of the following theorem of Laplace. 
(2.) Let V be the sum of the products of every attracting 
particle into the reciprocal of its distance from the attracted 
