MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
521 
the surface of the nucleus 
6-^— a\m?¥ {a,, a') 6—^, 
4/35rY, 
5 a' 
because 
/ (cos' =/, (cos' 6-\ =/, [cos' «- J( 1 -/')[ 
(«.) 
where f’ is a very small quantity, and where consequently f^f’ is a negligible pro- 
duct. 
Solving («.) with respect to j(3, we obtain 
^iY 2 =— /3A^cos^^— ( 16 .), ^yh—f,^{a^)-\-a\m'^{a„a'). . . . . ( 17 .) 
If we make Y^=^h and substitute the value of y in r, we shall have 
r=ai(l — |3/i cos^^). 
Hence ^h=e^, e, being the ellipticity of the surface of the nucleus, and consequently 
of the shell’s inner surface. 
The general form of the function upon which depends the attraction of the nucleus 
on an exterior point, may be found by substituting these values in (7.), having first 
changed into ^ and Wo, Wi, &c. into Y„ &c. ; hence 
V3 = 
(cos' afj g/ - 
(18.) 
assuming that the mass of the nucleus is equal to the mass of the solid spheroid 
included within its surface, having the same law of density as the shell. The truth 
of this assumption will appear more manifest further on. 
7. If, in a similar manner, we substitute the values of (BWg, &c., (3 Wg, &c. in (8.), 
then develope r in both (8.) and (18.), after the differentiation of (18.) we shall have 
the sum of the attractions of the nucleus and shell on a point at the shell’s outer 
surface, by adding the resulting expressions. If to this sum we add the term pro 
duced by the centrifugal force, we shall have for G, gravity at any point of the 
surface having 6 for the complement of its latitude, the expression 
G= G^( 1 -)-X cos^ ^), (19-) 
where 
3(/'(a)— ^,a?/(a.)) r«(l+^(«', «,)]«i+2e-3e,/*a?, . . (20.) 
all terms of the second order being as usual neglected, rriy standing for the value of 
m corresponding to and G' for gravity at the equator. When the mass is supposed 
5 
to be entirely fluid aj=a'=l, m,=m, q'(a')=0, [^^=1, and consequently 
as should be expected. The foregoing is the most general expression for the varia- 
tion of gravity at the earth’s surface yet obtained ; gravity at any point being expressed 
