€0 EEPORT — 1891. 



It is obvious that these lines and surfaces are always at right 

 angles the one to the other, for, in whatever direction there is 

 displacement, in that direction pressure must be changing. 



Also, if a number of lines of displacement be taken so as 

 to form a tube, there will be no displacement across the walls 

 of the tube : so, if S be the area of any section, and x the average 

 displacement across it, Sx = a constant, the tube being small. 



Such a tube can only, of course, proceed from a place where 

 the dielectric is pushed back from the conductor to a place 

 where it is drawn in ; and if Q = charge at one end of the tube, 

 — Q = charge at the other end. 



Suppose that the strain of the dielectric due to the charges 

 on any system of conductors is maiopcd out by drawing surfaces 

 of equal pressure in such a way that the pressure on any one 

 differs by unity from the pressure on either of the adjacent 

 surfaces, and by drawing tubes of displacement, so as to fill all 

 space, and each of such size that the flux (<S . x) along it is 

 unity. 



The dielectric is then divided into cells, each of which can 

 be shown to contain half a unit of energy. 



For consider the cell in 

 the figure.'-' Let a = distance 

 between two consecutive sur- 

 faces of equal pressure, s area 

 of section of the surface p by 

 the tube. 



Now, if a unit volume be 

 displaced a distance x, the 

 force of restitution is Ea:, 

 and the energy of displacement = -^ . Ex^. 

 So, if X = displacement across s, 



Energy = -J- . Eo;-* . a . .s. 

 But {p + l)s — ps = Ex'as, 



Or Bxa = 1. 

 Also, the tube is a unit tube ; 



.-. xs = 1 ; 

 .*. Energy in the cell -= -|. 



As long as the cell is of finite size this reasoning is incorrect, 

 though the result is true ; but, obviously, whatever the size 

 of the cell, it may be divided into a very large number of equal 

 small cells, say, a million. Then each of these contains one 

 two-millionth of a unit of energy. 



Hence, in any particular case, if we can count the cells, we 

 can find the energy of the system. Now, if a system of con- 

 ductors contain chai-ges Qi, Q2, Qg, and so on, they will 



* lu the figure, for^Jxl, reacl^-|-l. 



