578 
Mr. Airy on the figure of the earth. 
be satisfied, in consequence of the introduction of terms mul- 
tiplying f 6 , which would not destroy each other. It would 
be necessary then to assume for R a value of the form 
a 
{l + «• 1 - f*' -(4 + 4 ) • (*"- (*'*-(4- + 4 6|*'*+5 4 : 
by which there would be introduced two new functions of 
a, e , &c. in the general expression for V, and in that for the 
force at any point ; and one new function of a, e, See. in the 
expression for the force at the surface. But as a power of 
is introduced, which did not appear before 3 there would be 
found, by the process of Art. 14, one equation which was 
not found in the preceding approximation. By this equation 
the new function could be eliminated, and therefore the 
force at any point could be expressed in the same manner as 
in the preceding approximation, namely, by means of one 
function of (a) multiplying a quantity depending solely on 
the form of the external surface, and the latitude of the point 
on that surface. The same may be proved for every suc- 
ceeding approximation ; and thus we arrive at the following 
theorem : “ If a heterogeneous fluid, the particles of which 
attract each other with accelerating forces inversely "propor- 
tional to the squares of their distances, revolve round an 
axis ; and if the proportion of the centrifugal force at the 
equator to the whole force there be given ; the force at any 
point of the surface can be exactly expressed from a know- 
ledge of the form of the surface and the position of that 
point, without any knowledge of the law of the internal 
density.” This is likewise true if the interior be solid. 
G. B. A. 
