270 PROFESSOR G. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
Again the potential of S at P is Uq, and since F lies on the negative side of S' and 
has co-ordinates relatively to the n, s, t axes at P' given by 
11 = — T cos a, s — — r sin a, 1 = 0; 
since further the density on S' is negative, we have 
V 
clU, , , . dU 
r cos « ~ (—) + "J" “ 
dn 
ds 
Therefore 
V — V = r cos a 
dn ^ ’ dn^^’ 
= 47rr8 cos a. 
The differential with respect to n of the potential of S falls abruptly by TttS as we 
cross S normally from the negative to the positive side; and the differential of the 
potential of S' rises abruptly by the same amount as we jiass on across S'. It 
follows tliat dvjdn on the inside of S is continuous with its value on the outside of S'. 
The surface S to which this theorem is to he applied is a slightly deformed 
ellipsoid, and the curves are the intersection of the two quadrics confocal with the 
ellipsoid which is deformed. The curves start normally to the ellipsoid, and where 
they meet S the angle a will be proportional to the deformation whereby S is derived 
from the ellipsoid. It follows that cos a will only differ from unity by a term 
proportional to the square of the deformation, and as it is only necessary to retain 
terms of the order of tlie first power of the deformation, we may treat cos a as unity. 
We thus liave the result 
V — v' = 47rr§. 
Suppose the curve PP' produced both ways, and that il/g, are two points on it 
either botli on the same side or on opposite sides of the double layer. 
Let be equal to let ^ be measured in the same direction as n, and let ^ be 
a small quantity whose first power is to be retained in the residts. 
Let I’o, be the potential of the double layer at Mq and respectively. 
When { does not cut the layer we have 
and when it does cut the layer 
Vr 
— Vj = 477x8 — ^ 
dv 
dll " 
In the apjDlication which I shall make of this result the surface S' will actually be 
inside S. Then Vq will denote the potential at any point not lying in the infinitely 
small space between S and S', and is the potential at a point more towards the 
inside of the ellijjsoid by a distance ^ ; 8 is the surface density on the external surface 
