THE ELECTRIC AND LUMINIEEROHS MEDIUi\[. 
291 
the mechanical forcive may be readily calculated. If u denote the radial displace¬ 
ment, the normal and transverse principal tractions at any point in the spherical 
dielectric shell are 
P = 
(Jm 
\ch + 
clu 
'dr 
Q = x 
du . ^ u 
dr 7 
+ 2p 
u 
where p, X -f- fp are the moduli of rigidity and compressibility of the material. The 
electric force at any point is where H'C is the charge on a coating : hence (§ 36) 
the mechanical bodily force is derived from the potential — (K — l)/ 87 r . ; and 
there is also an outward normal traction over each coating equal to —K/ 877 .P'r“h 
The equation of equilibrium of a conical element of volume is 
dr 
) — 2 m = 
giving 
P. 
dr j 
K - 1 I 
K - 1 Tf_ 
'Itt (a, + 2yu,) 
and therefore 
4/i 
X + 6yu. (K — 1) 
P=(3X + 2riA-yB-^^'-i^^, Q = (3X + 2rtA + ^fB 
A.-2/4 (K- 1)1-^ 
X + 2/4 SttH 
The values of A and B are determined by the normal tractions at the coatings 
and are, when the coatings are wholly supported by the dielectric. 
= in and r = 
A = 
1 
i\r„ (rp -I- rp'o + r/) Stt (3X + 2/4) ’ 
B = 
rp’2 ^ ^ 327^/4 ’ 
where S represents 1 — (K — 1 ). 4p,/(X -}- 2 /i,). 
It will suffice to state the results for the case of a thin shell of radius a with 
adhering coatings : then A = .9^/2477 (3X + 2 p,) B = 
The coefficient of expansion of the radius of the sphere, due to the electric stress, 
is uja = (X -h 2p, — ju,K)/ 87 r/x (3X -j- 2|u). F^, and the coefficient of expansion of the 
volume of the sphere is three times this. It is easily verified that when the shell is 
thin the stress in the material of the dielectric is made up of a pressure KF^Stt 
normal to the shell combined with a pressure (K — 2 ) F^/ 87 r in all directions 
tangential to it.^ 
[76t. The circumstance that these results are independent of the radius of the 
sphere suggests an extension of their scope. Whatever be the form of the 
dielectric shell provided it is of uniform thickness, F will be the same all over it; 
and the mechanical force acting on its substance, being derived from a potential 
— (K — 1 ) F^Stt, will be directed at each point along the normal hi to the shell. 
Consider the internal equilibrium of an element of volume SS hi, of which the 
opposite faces SS are elements of level surfaces bounded by lines of curvature for 
which B^, Bj are the principal radii: it will be maintained if a pressure of intensity P, 
* This naturally differs from Kirchhoff’s result, ‘ Wied. Ann.’ 24, p. 52, § 4. 
t Rewritten December 2, 
