MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
515 
This expression shows that the parallels of mean pressure are symmetrically situated 
at each side of the equator. 
If for z its value be substituted, and the condition of equality of volumes be re- 
membered, we shall obtain 
cos = +- 
^A^+A,A,+A; 
(3.) 
Ti being- supposed to differ very little from Ai. 
If the changes in the oblateness of the nucleus be small, cos will be nearly con- 
stant, and the parallels of mean pressure will oscillate but slightly about their mean 
position. Hence if A 2 =Ai(l -j-s), s will be a very small quantity depending on the 
difference of the ellipticities of the two surfaces, but by Taylor’s theorem 
F(A2)=F(A)-f-£F'(A)-l-^F"(A, )+.... 
Hence if the surface of the shell should change with every change of form of the 
nucleus, s will be infinitely small, and consequently 
COS^i = d (4.) 
If the surface of the shell should not change its form for every change in form of 
the nucleus, s might yet be so small as to be negligible compared with other quanti- 
ties entering into our analytical expressions, and hence in such a case the above ex- 
pression for will be approximately true. 
As is the complement of the latitude at the parallel of mean pressure, we may 
conclude from the preceding investigation, that, at the parallel the square of the sine 
of the latitude of which is one-third, the pressure of the fluid is always the same as if 
the surface of the nucleus were one of equal pressure for the shell. 
Hence if we represent by f the centrifugal force at the equator of the shell’s outer 
surface, where the radius is supposed equal to unity, and by f its value corresponding 
to tti, equation (1.) becomes 
P=n-(/-/)(cos*^-^)y^e«^«. (5.) 
where a^ represents the radius of a sphere equal in volume to the spheroid included 
within the stratum of the nucleus with the radius r, and which consequently differs 
bjit little from r. 
The simple expressions for the pressure on any stratum of the nucleus and on the 
shell’s inner surface obtained in this section, will be found useful in the course of 
these researches, and particularly in the succeeding section. 
II. THE VARIATION OF GRAVITY AT THE EARTH’S SURFACE. 
4. It is in general evident, that if the laws of arrangement of the molecules com- 
posing the shell and nucleus be different, the variation of gravity at the surface of 
