378 Mr. Hennessy on the Connexion between the Rotation of 

 touches the a xis be called A', then a? = A' (1 — sin ■^), 

 R = ^r2 + a^sm^^^. Thus, if p be given as a function of « 

 or R, it can be expressed as a function of r and % or (r, ^). 

 y, a function of a?, can also be expressed as a function of "^ or 

 F (^), and (a) then becomes 



-y* ^ J^ 2 7r A'<^(r,^)^cos^rfr^>^. . (3.) 



In the case of a spheroid, where A' is the semi-polar axis, 

 and the plane of y, z coincides with the plane of the equator, 

 ■^ is evidently the latitude of the ring, and y = a (1 + ^) cos ^, 

 e representing the ellipticity of the generating curve. If the 

 earth be supposed to consist of an infinite number of sphe- 

 roidal shells, it has been shown that the density of any shell 

 which will agree best with the known ellipticity of the earth 

 and its mean density, is represented by the formula* 



A sin a a 

 ^ a 



where p represents the density of the shell, a its semi-polar 

 axis, and A and q constants. The value assigned to q is ex- 

 pressed by q — — i p representing the semi-polar axis of 



the entire spheroid. If I^ represent the moment of inertia of 

 the internal spheroid, then 



''=(l + ^)cos^' (*•) 



and 



b representing the semi-equatorial axis of the entire spheroid. 

 Butt 



\ tanqa/ \^j q''a-^ qatanqa 



2 2 9^ f^^ 



^ iauqa iaiVqa 



To completely eliminate e and a between this equation and 

 (4>.), is a step which must beat present considered impossible. 



An approximate solution could be obtained, but the result- 

 ing expression would be so long and complicated as to be 

 entirely useless. It is evident, however, that when e is small, 

 its equality in every shell can be assumed without material 

 error. Let e tlierefore be equal to the ellipticity of the exter- 

 nal shell, so that 



* Airy on the Figure of the Earth. f Ibid. 



5m 



