230 
PROCEEDINGS OF SECTION A. 
to.) If, now, a membrane immersed in the medium be submitted to a 
pressure P, the further amount of energy stored up is ^ l P a , the same as 
if the cavity (r is very small) did not exist, for there is no occasion 
given here (as in the former case) for another pressure to do work. 
Thus the energy of the medium is i l P 2 + \ E But suppose the 
order of displacement to be reversed. The displacement of the mem- 
brane will cause an amount of energy -§• l P 3 to be stored up in the 
medium. Also the displacement will cause variations of pressure 
in all parts of the medium. (A uniform pressure throughout 
the medium does not affect matters at all, and may be supposed 
to exist if it be felt a difficulty that without it the dis- 
placement of the membrane will cause negative pressures on 
one side of it.) Let the pressure so caused in the neighbour- 
hood of the cavity be p. If, now, the quantity q be introduced, 
it has to be forced into the cavity, not merely against the stress its 
own presence causes, but also against p. Thus the work done in 
forcing the liquid in is p.q + Whilst this is being done 
the membrane is pushed back a certain amount. This bears to q the 
ratio that the solid angle subtended by the membrane at the cavity 
does to 47r, say, This displacement takes place against P, and 
therefore the medium is deprived of energy to the amount P<? — . 
Thus the total energy = \lV 2 +pq + ^ E — • P q ~ . 
This must equal the value obtained before, and therefore p = P . 
Thus the surfaces of equal pressure in the neighbourhood of a dis- 
placed membrane are the loci of points at which the membrane subtends 
equal solid angles. The lines of displacement are perpendicular to the 
surfaces. 
These proofs and results have their exact analogues in electro- 
magnetism. The displacement of the incompressible medium corresponds 
to magnetic induction ; the incompressibility represents the fact that 
lines of induction must return into themselves, or, in other words, 
that the number that enter any closed space is the same as the number 
that leave. Just as the displacement in any direction is proportional 
to the rate of change of pressure in that direction, so the induction in 
any direction is proportional to the rate of change of magnetomotive 
force in that direction, the constant of proportion being the 
permeability. When a current runs round a circuit it exercises a 
uniform magnetomotive force over every unit of area of a surface 
(any one) drawn to have the circuit as contour. 
Thus the results obtained above must have their exact analogues 
in magnetism, which may be proved at once from the corresponding 
hypothesis. I have only used the strain theory because it gives such 
good mental pictures. 
