Scientific Papers - Vi

1916] APPROXIMATE SPHERES AND CYLINDERS 385
cubes of C being neglected at this stage. On introduction of the value of Hn from (8) and of Su from (5),
</>i = — H6 log (w0&) + ^HQ {3Cf12+ 5<722 + 7032 + ...}.........(10)
Thus <^i/Q = 2 log('«06)-|-{3(712 + 5(722 + 7C'31!= ...}...........(11)
In the application to an electrified liquid considered in my former paper, it must be remembered that ICQ is not constant during the deformation. If the liquid is incompressible, it is the volume, or in the present case the sectional area (cr), which remains constant. Now
_ [^ dd Zcr =
o vito T on) u0- j (
= 27T
so that if a denote the radius of the circle whose area is cr, i jp
Accordingly,
and (11) becomes
hlQ = 2log(bla)-Cf-Mf-...-(p-l)Op*, ......... (13)
the term in Ol disappearing, as was to be expected.
The potential energy of the charge is ^faQ. If the change of potential energy due to the deformation be called P', we have
in agreement with my former results.
There are so few forms of surface for which the electric capacity can be calculated that it seems worth while to pursue the approximation beyond that attained in (11), supposing, however, that all the e's vanish, everything being symmetrical about the line 0 = 0. Thus from (4), as an extension of (7) with inclusion of O2,
E. VI.
ff
(15)
WF,+ ...), ............(16)
25
or with use of (8)
/•27T /J/3iv. p. 184 (1882) ; Scientific Papers, Vol. n. p. 130. t Maxwell's Electricity, § 126.t with the walls of the tube is of unit specific resistance.