FIGURE OF EQUILIBRIUM OF A ROTATING MASS OF LIQUID. 
2G9 
curves, and let a be tlie angle between the curve and the outward normal at any 
point. At every point of S measure along tlie curve an infinitesimal arc t, and let r 
be a function of the two co-ordinates which determine position on S. The extremities 
of these arcs define a second surface S\ and every element of area dcr (jf S has its 
corresponding element da on S'. Suppose that *S is coated with surface density 8, and 
that S' is coated with surface density — S', where 8 da — 8' da'. The system SS' 
may then be called a double layer, and its total mass is zero. We are to discuss the 
potential of such a system. 
Let and U ( —) be the external and internal potentials of density 8 on 
and Uq their common value at a point P of S. At P take a system of rectangular 
axes,-n being along the outward normal, and s and t mutually at right angles in the 
tangent plane. 
In the neighbourhood of P 
r-r / . \ TT- . dt/, . , dU , , . , dU , , . 
ds 
dt 
TT I \ rr . d.U, , . dU , . , dU, , 
t/(-)= + „_(_) +s-(-) + i--(-). . . 
ds 
dt 
In the first of these n is necessarily positive, in the second negative. 
dU 
ds '' ' ' ds '' ' ds 
Now ^^( + ) = (—) = ~ ; and the like holds for the difterentials with respect 
to t. 
Also by Poisson’s ecpiation 
Let PP' be one of the family of curves whereby the double layer is defined, and 
let P' lie on S', so that PP' is r. By the definition of a the normal elevation of S' 
above S is t cos a. 
Let V, v' be the potentials of the double layer at P and at P'. 
The potential of >S" at P' differs Infinitely little in magnitude, but is of the opposite 
sign from that of S at P ; it is therefore — The 2 )oint P' lies on the positive 
side of S at a point whose co-ordinates may be taken to be 
n = T cos a, s = T sin a, t — 0. 
Therefore the potential of S at P' is 
TT , dU. , , . 
-f T cos a — (+ ) -t- 
T sin a 
dU 
ds * 
dU 
v' = T cos a ( + ) + T sili “ 
d)i ' ' 
dU 
ds 
Therefore 
