466 F. E, Nipher—Kapression of Electrical. Resistance. 
in energy being due to work done on the sphere by some 
external source, causing the sphere to collapse. If the element 
ds sweeps through a distance dr, the stored energy will be 
’ dE=dF dr (1) 
in which both dF and dr are essentially negative. 
Substituting in (1) the above value of dF’ and remembering 
a 
that are 
and ds = r°dw, 
where dw is the solid angle subtended by the element ds, we 
have dE = @ i d@, 
87 Tr 
or E'~ka@ [fF de, 
8x r 
where one integration is carried over the surface of the sphere, 
and the other is carried inward between the limits r and”. 
Performing the integrations, we have 
ni-E=€(—-+) (2) 
r r 
Lg 
But 5 s is the energy of a sphere of radius 7’, charged with 
@ units of electricity, and hence the potential of the sphere on 
itself between the limits 7 and 7’ is equal to the difference in 1ts 
initial and final energy. ; 
If the sphere were connected with the ground by a wire of 
resistance (/2), the radius (r) might be changed in such a man- 
ner as to preserve the potential (V) constant. In this case 4 
current of constant intensity would flow through the wire, and 
as V= 2 it is clear that + must change at a uniform rate, or 
as , 
Pe ee (3) 
v 
where ¢’—t is the duration of the operation. Further, 
4 2 
Fae Ss ee eodern 
Tt r 
hence dE=dF dr=20 p ds dra dr dw, 
