60 Mr. J. L armor. Theory of Electrodynamics, [June 2, 



Suppose that the distance between the plates, with no air bubble 

 introduced, is slowly increased from c to c + dc against the tension 

 T 2 which tends to draw them together. The work done against T 3 

 in raising the upper plate is the only source of the additional energy 

 of the system which appears when that separation is effected. Now 

 the value of F is not thereby altered, as the electrification remains 

 constant. The volume energy contained in a cylinder of the liquid 

 of unit sectional area is therefore changed from C 2 F 2 c to C 2 F 2 c (l + Sc/c). 



d2 2 

 The corresponding surface energy is changed from 2 2 to 2 2 -| c. 



7^71 



Therefore, by the principle of work, T 2 = C 2 F 3 -f -^-. 



Now suppose a large air bubble introduced between the plates ; 

 and suppose its volume to be increased by Sv owing to a virtual dis- 

 placement produced by pressing in more air. The virtual work Py 

 so done must be equal to the increase of the internal energy of the 

 system due to the displacement. This increase may be calculated 

 from the energy function of the actual conformation, not of the dis- 

 placed position, as internal equilibrium subsists ; and all considera- 

 tions as to change of intrinsic energy may thus be evaded.* For a 

 bubble of small dimensions its surface would, in fact, be increased by 

 the supposed displacement, and so there would be an increase of 

 intrinsic capillary energy; with surface tension T, radius a, and a 

 semicircular meniscus TTC, there would be an increase of amount 

 2 7r a CToa, that is vrcSv/a. Again, the electric forces FI and F 2 in the two 

 media, measured not close up to the meniscus, are equal, because the 

 plates are each at uniform potential. Close to the centre of the 

 meniscus they are both tangential to it, and must also be equal on the 

 two sides. But along the slope of the meniscus they are oblique to 

 it and are unequal, their relation being determined on the ordinary 

 theory by the continuity of the normal component induction and of 

 the tangential component force. Thus the difference of electric 

 pressure across the meniscus will vary from point to point of it, and 

 its form will, therefore, be slightly altered by the electrification. It 

 follows from the observations of Quincke and others that its capillary 

 constant will not thereby be altered, as was to be expected because 

 there is no extra supply of molecular surface energy. There is also 

 the alteration of intrinsic energy due to the fact that the expansion 

 of the air bubble alters the electric force. But, according to the 

 principles just stated, these changes of intrinsic energy balance each 

 other, because all the parts of the medium are in internal equilibrium. 

 We may therefore consider the annular mass between two ideal 

 coaxial cylindric surfaces at a distance from the meniscus, one in the 



* For applications of this principle, cf. Helmholtz, ' Wied. Ann.,' vol. 13, p. 388; 

 and Kirchhoff, * Wied. Ann.,' vol. 24, p. 57. 



