272 PROFESSOR O. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
The density + pe [1 — {s — 0] on t, together with — pe [1 — Xe (s — t — d^)] dt 
on t + dt, constitute an infinitesimal double layer; and since the positive density on 
each t surface niay be coupled with the negative density on the next interior surface, 
the finite double layer may be built up from a number of infinitesimal double layei’S. 
d~ 
Hence dt dt is the excess of the potential at F above that at Q of an 
infinitesimal double layer of thickness edt, nud with surface density pe [I — Xe [s — f)] dt 
on its exterior surface. 
di 
We may now apj^ly the result Vq — = 47 rST or 0 , according as { does or 
does not cut the double layer, and it is clear that 
Tp' V -FF 
' dii_ 
47rpe" [l — Xe (.s — f)] or 0, 
according as 2 is greater or less than f. 
In the next j^lace, wm must integrate this from t = s to ^ = 0 , and the result will 
have two forms. 
First, suppose z > s; then for all the values of d ~ > t, and the first alternative 
holds good. Therefore 
d 
ds 
V — V 
^ 0 ' ~ 
47rpe' [.s — ^Xei'*] . 
Secondly, sup})ose s < ; then from t = s to t = z, z t and the second alternative 
holds, while from t = z to t — 0, z'y> t and tlie first holds. Therefore 
We have now to integrate an’ain from .9 = 1 to 5 = 0 . 
o o 
From s = 1 to s = 0 < s and the second form is ap})licahle ; from s = z to = 0 , 
s > ,9 and the first form applies. 
Therefore 
d V f ^ r- 
— F. — 62 = Ivrpe^ j [2 — Xe (s 2 — -^ 2 :")] ds + 47rpe” | [s — o-Xe.s”] ds 
= 47 rpe~ ]2 (1 — 2) — Xe [^2 (l — 2") — ^z- (l — 2)] + ^2^ — IXez^} 
= 217pe~ [2z — 2' —- Xe(2 — 2” + i‘ 2 :®)}- 
Finally, we have to multiply — « ( Fq — Id) by an element of negative mass at the 
jjolnt defined by 2 and integrate throughout R. The physical meaning of this 
integral will be considered subsequently. 
We have already seen tliat such an element of mass is given by 
