THE FORCE ON A SMALL CHARGED BODY 69 



suggested by, and most appropriate to, the old "action at a 

 distance " view of electric action, and it might appear out of place 

 with our new conception of action through the medium. But if, 

 as in the text, we only say that the strain is the same, however 

 produced, as if the direct action occurred, the method is perfectly 

 justifiable for purposes of calculation, and it is the best in the 

 first treatment of the subject owing to the ease with which it 

 admits of mathematical representation. As an illustration of this 

 method, consider the somewhat similar case of the equilibrium of 

 a body pulled by a number of strings, with given forces applied at 

 the free ends. We consider these forces as applied directly at the 

 points of attachment of the strings, and do not trouble to consider 

 how the strings behave or to take them into account, so long as we 

 are dealing only with the equilibrium of the body. For experi- 

 ment has shown us that if a force is applied at one end of a string 

 an equal force is exerted by the string at the other, and we there- 

 fore are justified in a kind of " direct action" treatment, though 

 obviously the string has an all-important part to play, and 

 must be considered if we are to have a complete investigation 

 of all the phenomena accompanying the equilibrium of the 

 body. 



There is a method of treating electrical actions in which the 

 mathematical representation fits in perfectly with the supposition 

 of action by and through the medium. It is fully set forth in 

 Maxwell's Electricity mul Afn^n, ti\tn. vol. i, chap, iv, and we shall 

 here only point out verv briefly the principle of the method for 

 the sake of advanced readers who may wish to compare it with the 

 easier, though much less general, method which starts from the 

 inverse square law. 



Let us suppose that we \\i-li to find how electrification will 

 be distributed under given conditions on any given system of 

 conductors in air. Starting with the idea of the electric field 

 as a space in which electric .strain and intensity may be mani- 

 fested, we must deduce from experiment that in air the strain has 



v where the same direction as, and is proportional to, the 

 intensity, and that we have chosen units so that E = 4<7rD. 

 Then introducing the idea of the energy possessed by the system, 

 we must show from our experiments that its amount depends solely 

 on the configuration. From this it follows that a potential 

 V exists, and the intensity at any point may be expressed in 

 NTIIIS of V. 



The total energy of the system may be shown to be 



where T is UK- ui nsity on the element of surface dS, the 



integral being taken over all the conducting surfaces. 



