398 Royal Society. 
constant independent of A and &. The following propositions are 
then demonstrated :— 
The potential, on a given external point, of a homogeneous solid 
bounded by the surface (/, 4), varies as the mass of the solid, if 4 
vary while / remains constant. 
The potentials, on a given external point, of the homogeneous 
shells (4. i) (* 4) are proportional to the masses of the shells. 
1 1 
The homogeneous shell ¢ ?) exercises no attraction on an in- 
terior mass. j 
The external equipotential surfaces of the homogeneous infinite- 
simal shell G ated) are the surfaces (h, 4), in which / is arbi- 
trary and & invariable *. Pew 
The potential of the homogeneous infinitesimal shell (4 h ) 
upon an exterior point, is 
4a he dh 
— dkwl(h —_ 
n v ( ) h yp (A) 
and upon an interior point, is 
Fak y(n) fe. : 
n » VW) 
= focnan et 
(In these expressions (/) is e” , and h at the lower limit 
in the first, is the parameter of the surface (/, #) which passes through 
the attracted point. The density of the shell is supposed to be unity. ) 
' 
The potential of the finite homogeneous shell { 4,, i! ) (density 
=1) upon an exterior point (&, , Z), is 
Ar. nf? dhs egy (he dhe, gees 
Suh) if. Va) uf Hay Su Way S | 
in this expression it has been assumed (for simplicity) that 4,, is inde- 
pendent of &. Also h', A! are the values of A corresponding to &", k’, 
when / and & vary subject to the relation f (é,n,¢,A,4)=0; and /, in 
the last integral, is the function of f, é, n, £ determined by this relation. 
The differential equations (2) are satisfied in the case of the ellip- 
soid, For if we put its equation in the form 
a = 
it is evident on inspection that 
Ph Ph, Ch ( 1 1 1 ) 
tap ANGER PL EA 
and dk\2 dk\2 dk\* dk _ 
(a) +(%) +z) +4307 ° 
In this case we find U(h)=((A+h)(B +h) +h) )t, and the 
above general expressions lead to the known results. 
* It is known that the last two propositions imply the first two (see Mr. 
Cayley’s “‘ Note on the Theory of Attraction,’’ Quarterly Journal of Mathematics, 
vol. ii. p. 338) ; though this is not the order of proof in the present paper. 
