607] 
A MEMOIR ON PREPOTENTIALS. 
323 
q = Positive. Art. Nos. 3 to 7. 
3. I consider first the case where q is positive. The value is here 
«2 
2 (r|) s 1 ( r^.? ffi 2 x q ~ x dx ) 
~ P lW) 2^ (T(^s + q) ~Jo (T+xji*+v} ; 
or, since is indefinitely small, the «-integral may be neglected, and the value is 
, 1 a (nr rg 
« 2q P r s + q) ' 
Observe that this value is independent of R, and that the expression is thus the 
same as if (instead of the disk) we had taken the whole of the infinite tangent-plane, 
the density at every point thereof being = p. It is proper to remark that the neglected 
terms are of the orders 
so that the complete value multiplied by « 2? is equal to the constant p 
myr q 
r (hs + q) 
+ terms 
of the orders 
4. Let us now consider the prepotential of the remaining portion of the surface; 
every part thereof is at a distance from P exceeding, in fact far exceeding, R; so 
that imagining the whole mass j" p dS to be collected at the distance R, the pre 
potential of the remaining portion of the surface is less than 
j pdS 
R s+2q ’ 
viz. we have thus, in the case where the mass j p dS is finite, a superior limit to the 
prepotential of the remaining portion of the surface. This will be indefinitely small 
in comparison with the prepotential of the disk, provided only « 2? is indefinitely small 
i+^- 
compared with R s+2q , that is, « indefinitely small in comparison with R 2q . The proof 
assumes that the mass j pdS is finite; but considering the very rough manner in which 
jpdS 
the limit -p-v-: was obtained, it can scarcely be doubted that, if not universally, at 
least for very general laws of distribution, even when Jp dS is infinite, the same thing 
is true; viz. that by taking « sufficiently small in regard to R, we can make the 
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