Flow of Electricity in a uniform plane conducting Surface. 385 



Integrating (a) with respect to t, we get, as before, the equation 

 of vis viva ; this mode of deriving which enables us to see the 

 error committed in supposing that, when the equation is formed 

 for a continuous mass, the equal and opposite forces acting at 

 the common boundary of two elements will be multiplied by the 

 same velocity, viz. that of the bounding surface. 



It is clear that we must treat the motion of each element ex- 

 actly as if it were collected at its centre of gravity — that the 

 velocities by which we multiply the forces acting upon it are not 

 the velocities of their actual points of application, but that of 

 their hypothetical point of application, viz. the centre of gravity 

 of the element — in other words, that the velocity by which we 

 multiply must be the average velocity prevailing throughout the 

 element ; from which it follows that the force p, for example, 

 when regarded as acting on the element originally considered, 

 must be multiplied by the velocity v ; but when it is regarded 

 as acting on the element immediately in the rear of the former, 



it must be multiplied by v_i or v—-r- dec; so that, instead of 



GtV 



those equal and opposite forces disappearing from the equation 



dv 

 of vis viva, they will give rise to a residuum, viz. p-j-dx; and the 



sum of these small terms taken for the entire wave will in general 

 be a finite quantity. 



It can hardly be necessary that I should point out the bearing 

 which the foregoing conclusions have upon the applicability, 

 when compressible bodies are dealt with, of the principle of 

 virtual velocities. 



6 New Square, Lincoln's Inn, 

 March 3, 1875. 



XLI V. On the Flow of Electricity in a uniform plane conducting 

 Surface. — Part I. By G. Carey Foster, F.R.S., and Oliver 

 • J. Lodge*. 



[With a Plate.] 

 1. rriHE objects of the following paper are : — first, to show that 

 -*- the most important laws relating to the flow of electri- 

 city in a plane conducting sheet of uniform conductivity can be 

 established by mathematical considerations of such a simple 

 kind that there is no reason why they should not be introduced 

 into ordinary teaching, as well as into the more complete ele- 

 mentary text-books of electricity ; and in the second place, to 

 describe methods that may conveniently be employed for testing 



* Read before the Physical Society, February 27, 1875. Communi- 

 cated by the Society. 



