6 
PEOFESSOE DONKIN ON THE ATTEACTION OF SOLIDS BOUNDED 
When the thickness of the shells is infinitesimal, this proposition may be enunciated as 
Theoeem III. The potential, u^jon a given external mass, of the homogeneous shell 
^h, varies as the mass of the shell, ifh. vary while k remains constant. 
6. The above conclusions were deduced from equation (2.), art. 2. Let us now take 
equation (3.) and apply it in a similar manner, taking for the parameter &, no longer Tc, 
but an indeterminate function of h (not containing k), say 
This gives, without ambiguity of sign, 
Q,=f{h)Q, 
(for by the convention made, art. 1, as to the signs of Q*, &c., and Q* must have the 
same sign or not, according asy’'(7i) is positive or negative). 
Hence, writing the left-hand side of (3.) in full, and introducing the second of the 
conditions (6.), we have 
w/'(/? 2 )|^y^ — -w/'(/q) J + (terms involving 
Now let M be the potential of a given mass M exterior to both the surfaces {h^, k), {h^, k), 
r Cud(T~Y dY 
so that J)^u is =0 throughout the integrations. The integral | is evidently 
if V be the potential on M of a homogeneous solid (density =1) bounded by the surface 
dY 
(h, k)-, and by equation (8.), art. 4, ^=F'(^). ^'(A). 
The above equation thus becomes 
nY{k){f’{\)-^{h^)-f{h,)-^{h,)] =[^^^iiWf{h).dxdydzfj,\ 
The function f{li) has been so far arbitrary. Let us now determine it in such a manner 
that/''(A)-v^(A)= a constant independent of h and k ; or 
/W=a+bJA 
(A, B being two such arbitrary constants). Then the left-hand side of the equation 
vanishes ; therefore the right-hand side vanishes also, or 
• dxdydzfi-= 0 ; 
but since u is the potential of an arbitrary mass, this cannot be unless D^(A)=0. We 
may therefore (introducing the value of -^(h), art. 4, and including the arbitrary con- 
stants under the integral signs) enunciate 
Theoeem IV. If f(h) he defined hy the eguation 
then f(h) satisfies the eguation 
Dy(/t)=o. 
