ENERGY OE TIIE ELECTRO-MAGNETIC FIELD. 
229 
Suppose the existence of an infinite incompressible medium, each 
particle of which requires to displace it from its natural position a 
force proportional to the displacement. The constant of this propor- 
tion may be different in different parts of the medium. If its value 
be E, and a unit volume be displaced a distance cl, the force required 
is E d, and the energy stored up in this portion of the medium is 
i E cP* 
Suppose a membrane placed in this medium, the medium being 
uniform and of elasticity (everywhere) E. If a uniform pressure be 
brought to bear on the membrane, it will be displaced. Most of the 
displacement will take place at the edges, and the membrane will take 
a convex form to the pressure, the opposite to the form taken by a 
sail before the wind. The lines of displacement will run from front 
to back of the membrane, and will occur in greatest numbers, if 
number be made to represent amount of displacement, near the edge. 
In fact, in the case of a circular membrane, the disposition of the lines 
is given in Maxwell’s Electricity and Magnetism, Plate XVIII. 
Lines due to a circular current. 
If the pressure per unit area be P, the total displacement of the 
membrane, or volume moved through, will be proportional to P, say, 
IP, where l is a constant depending on the shape (really contour 
only) of the membrane and the elasticity of the surrounding medium. 
The energy stored up in the medium will be J IP 2 . 
Suppose, now, a second membrane placed in the medium, and the 
medium strained by a pressure P' on this membrane. The work done 
in straining it will be exactly the same as if there were not already a 
strain in the medium due to the pressure on the first membrane, 
because every particle requires the same force to displace it — a given 
amount in a given direction, uo matter whether it is already displaced 
or not. If, then, the total displacement of the second membrane is 
nP', the work done in submitting it to a pressure P' is i nP' 2 . 
But whilst the second membrane is being displaced the first 
membrane is displaced also, and the amount of the displacement is 
mP', where m is a constant depending on the nature of the medium 
and the shapes and relative positions of the membranes. A pressure 
P is acting on the first membrane all the time it is undergoing the 
further displacement, so that an amount of energy P. mP' is done on 
the medium thereby. Thus the whole energy stored up in the 
medium is ^ ZP 3 -bwPP'-|-i ?*P' 3 . Plainly the order in which the 
displacements of the membranes are supposed to take place could be 
reversed without altering the ultimate amount of energy stored up in 
the medium, so that the constant m may mean either the total 
displacement of the first membrane due to unit pressure on the 
second, or vice versd, The energy in the case of any number of 
membranes is in the same way easily proved to be i 2 l P 2 + 2 m PP'. 
Consider now another case. If a small spherical cavity of radius r 
exist in the medium, and a small quantity of incompressible liquid q 
be forced into it, the consequent energy of strain of the medium is 
where r is the radius of the cavity. 
(See paper already referred 
* See paper on Elastic Medium method of treating Electrostatic Theorems ; read 
before Section A, New Zealand meeting. Phil. Mag., vol. 34, p. 18. 
