8 
PROFESSOE DONKIN ON THE ATTRACTION OF SOLIDS BOUNDED 
Suppose now that the mass M is unity, and is concentrated at a point between the two 
surfaces, then Mi = 0, M 2 =l, and (9.) becomes 
V,-V.=i^(/(A)-/(A,)), 
where h is the parameter of the surface which passes through the point. 
Let the value of h which makes the surface (A, Jc) extend to infinity be denoted by A» ; 
then putting 7^2= Aoo, we have V2=0 (art. 7), and the last equation gives 
and again, if this value of V, be substituted, the same equation gives 
here is the potential of an infinitesimal shell on an exterior point, and V 2 on an inte- 
rior point. These expressions suppose that the densities of the shells are fW) ; 
hence, changing the densities to unity, and to Aj in the second, we obtain the follow- 
ing results : — 
The potential of the infinitesimal shell 
point is 
^A„ (density =1) upon an exterior 
and upon an interior point, it is 
B 
Now /'(A) (art. 6) is ; hence 
the above expressions 
and for an interior point, 
become, for an exterior point, 
(E.) 
(I-) 
9. In the expression (E.), the value of A at the lower limit of the integral is the para- 
meter of the surface which passes through the attracted point, and the potential has 
therefore the same value at all points of that surface ; hence 
Theoeem V. The external equijpotential surfaces of the homogeneous infinitesimal shell 
k-4-dk\ 
k ) 
are the surfaces (h, k), in which h is arbitrary and k invariable. 
The expression (I.) is independent of the position of the (interior) attracted point ; 
hence 
Theorem VI. The homogeneous infinitesimal shell ^h„ exercises no force cm an 
interior mass. It follows evidently that the homogeneous finite shell ^A,, possesses 
the same property. 
