wi) 
circuit due to unit pressure on the second. Thus during the dis- 
placement of the second membrane the pressure, 47C, on the first 
does a fresh amount of work, aS An, i.e. MCC’. Thus the 
total energy of the field is 
4 LC? + MCC’ +4UC??, 
which is the well-known important theorem. 
This can be extended to cases in which there are any number 
of circuits, by taking the corresponding membranes one by one. 
It is evident from the symmetry of this equation that the dis- 
placement of any membrane B caused by unit pressure acting on 
a membrane A is equal to the displacement of A caused by unit 
pressure on B. 
We may employ this last principle to find the displacement at 
any point due toa membrane under uniform pressure, corres- 
ponding toa closed circuit. Suppose we require the displace- 
ment ata point P in the direction x Let A bea very small 
area placed at P perpendicular to z. Then the displacement of 
A due to unit pressure on the membrane is equal to the 
displacement of the membrane due to unit pressure on a small 
membrane coinciding with A. 
Suppose a number of membranes placed at regular distances 
behind A, all perpendicular to x and equal to A; let there be » 
of them per unit length, and let the series reach to a point B 
very far from A. Let unit pressure act on each of these. Then 
the matter in which the membranes are imbedded will be dis- 
placed like a solid cylinder, and if d be the displacement of each, 
+ 
= the elasticity of the medium. 
An = ues 
pL 
eet 
Ar 
The effect of the displaced cylinder will be the same as if at A 
‘ Apen 2 ; 
there were a total displacement, i a spreading out uniformly 
in all directions, and at B a corresponding inwards displacement. 
The lines of displacement from A will therefore be practically 
straight lines ; and the amount of displacement of the original 
ae Q 
membrane is A pn 1622 where Q is the solid angle subtended by 
it at A. 
The displacement of the original membrane due to the pressure 
; : Q A pn 
on A is equal to the difference between A pn 16-2 and 6c? 
