518 
MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
From the values attributed to a! and will disappear, and from the origin chosen 
for the coordinates Wj will also disappear*. 
Let M represent the mass of the greater spheroid, I its moment of inertia with re- 
spect to its axis of rotation, Mj and Ij being the corresponding quantities relative to 
the smaller spheroid, then 
g=M^+^^||iw,+c.w.+...}-^4{|i.w,+c;w.+...}, 
Cg and 4 being constant coefficients. 
The artifice already employed may be used here in order to find expressions foi- 
jSWg, /S'Wg, &c. In this case the equation of equilibrium of the fluid at the surface 
of the greater spheroid is 
C=^[n-^[|^W,+...]}-r«;y(cos»0-|), 
M' and I' standing for the mass and moment of inertia respectively of the whole mass 
composed of the fluid stratum and solid spheroid. This becomes, on developing r, 
M' 
and making f, 
a » 
r3 
2 e® 
But also 
«;=W,-fW3-l-W,4-....-fW,. 
The truth of these two simultaneous expressions requires that 
3 W 2 I' a^f / 2/1 
M'“'2M^(^°® ^—3/ ' 
2 Ma^ 
But a being a number depending on the internal constitution of the sphe- 
fa!'^ 
roid, and'^=m nearly ; hence neglecting very small quantities, or making a\ = l, 
f3W2=. 
5 cm 
COS^^ 
( 12 .) 
2(5(r-3)' 
And in a similar manner we may obtain 

M 
/A, being used for brevity to represent and being a number analogous to <r. Now 
if r be developed in (8.), and all terms of the order (3^ be neglected, it will become, 
remembering that r'=a'(l+/3y), y being a function of the polar coordinates of the 
point. 
— m|i ;*)« 2 ( 5 ir — 3 )(* Si,— 3)] s}’ ' ' 
* See the work already cited, No. 21. 
( 14 .) 
