496 
MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
2. Let the earth be considered to have been originally a heterogeneous fluid mass, 
possessing only such general properties as those which have been established for 
fluids. Let x, y, z be the rectangular coordinates of (/j, an elementary molecule of 
the fluid ; and let the components of the forces, parallel respectively to the axes of 
X, y, z, be X, Y, Z, if the mass were not in rotation but at rest and in equilibrium. 
Let the mass rotate about the axis of 2 with an angular velocity at unity of distance 
represented by a, let the density at [a be represented by §, a function of x,y, z\ and 
let p represent the entire pressure upon pj, then* 
dp—^(^dx-\-Ydy-\-7jdz)-Ya^{xdx-\-ydy). 
The equation of the exterior surface of the fluid, and of the surfaces of equilibrium in 
its interior, will be 
'Kdx-\-Ydy-\-7jdz-\-cd{xdx-\-ydy) — 0. 
Let M represent the sum of the moments by which the rotation of the fluid mass 
was at first produced, and let the axis of 2 : pass through the centre of gravity, then-f- 
M 
The numerator at the right side of the above equation is evidently constant, and the 
denominator is a quantity depending upon the distribution of the molecules of the 
fluid about the centre of gravity of the entire mass. Let I represent the value of 
when a was the angular velocity of rotation of the body, and let I' be its 
value corresponding to an angular velocity of rotation denoted by a!. If a should not 
be known independently, but if I, I' and a! be known, then the equation to the free 
surface of the fluid can be transformed into 
'Kdx-\-Ydy-^7jdz-\-—^{xdx-\-ydy) — 0 ( 1 .) 
In any solutions heretofore given of the problem of the determination of the figure 
ps 
of the earth, the coefficient p seems not to have been noticed from the nature of the 
assumption which has been mentioned in (Art. 1.). 
When X, Y, Z are the components produced by the mutual attractions of the 
molecules of the fluid according to a certain function of x, y, z, then the values of 
X, Y, Z will in general depend upon the form of the entire mass and of its surfaces 
of equilibrium, and again that form will depend upon the values of the components 
and of a'. As the value of I depends upon the form of the mass, it must also depend 
upon X, Y, Z ; but it may be considered independently if the conditions upon which 
any particular solution of equation (1.) is to be made are such as to assign what func- 
tion it may be of the coordinates. Considered under this point of view, the three 
first terms, and the fourth or last term at the left side of the equation (1.), may be 
in 
considered as independent quantities whose values maybe found separately, and p may 
* Poisson, Mecanique, tom. ii. p. 536, edit. t Ibid. p. 79. 
