FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
277 
^ o) -Ipi. (j^o) 
1 
dVr, 
1 -f /3\^ 
^ 1 + /3 
Hence at the surface of the ellipsoid 
Ve — Fj = S ivh-^Si^ = 477x3. 
But this is the law found in § 9 for the cliange of potential in crossing a double 
layer, and hence Vg, Vi are the external and internal potentials of the double layer 
t8. 
d 2^ d 
dn k~ vdv ’ 
Since 
dn dn 7 ^ ’ 
dm 
I + ^\i diBf .. 
1— /D ) '9'" —j -y- o,' 
1 — pj O-Vq dvQ 
(25). 
Tliis result will hold good to the first order of small cpiantities if the surface be a 
slightly deformed ellipsoid, such as was the surface defined by t in § 9. 
In the elementary double layer t the density was pe [1 — \e {s — ^)] dt, and 
the thickness was edt, so that the thickness multiplied by the density was 
pe^ [l — \e {s — tj] dt dt. Since, however, we only need this to the first order, we 
may take it as pd'dtdt. It will now he convenient to change the meaning of /// to 
some extent, and to write 
e" = N hfSf. 
0 
Thus for the elementary double layer we have 
00 
T S = p di dt S hi’Si\ 
0 
d V 
It follows that in applying the formula (25) to determine ^ for the doulfie 
illlj 
system D, we may say that 
d~ d V ” 4:7rpip 
dsdt dn 
Id 
(v - !)■ (V — V’ - . -r~ S,: 
dvi) dvf^ 
Since the right-hand side does not contain g we have only to consider the integral 
f f ds dt = [ s ds = h . 
J 0 j 0 Jo 
Thus, for the system L), 
dV_ 
dn 
CO ^ 
0 
TrPdP 
Fl'n 
(v - 1)* 
i+_iv J 
1-is) ‘ dv„ ' ‘ 
(26). 
