336 Mr. 0. Heaviside on the 



the second system. Bearing this in mind, we can easily form 

 the corresponding formula in a less general case. Suppose, 

 for example, we have two fine wire terminals, a and b, that 

 are joined through any electromagnetic and electrostatic 

 combination which does not contain impressed forces, nor 

 receive energy from without except by means of the current, 

 say C, entering it at a and leaving it at b. Let also V be the 

 excess of the potential of a over that of b. Then'VC is the 

 energy-current, or the amount of energy added per second to 

 the combination through the terminal connections with, ne- 

 cessarily, some other combination. (In the previous thick- 

 letter vector investigation V was the symbol of vector product. 

 There will, however, be no confusion with the following use 

 of V, as in Part II., to express the line-integral of an electric 

 force. One of the awkward things about the notation in Prof. 

 Tait's ' Quaternions ' is the employment of a number of most 

 useful letters, as S,.T, U, V, wanted for other purposes, as 

 mere symbols of operations, putting another barrier in the 

 way of practically combining vector methods with ordinary 

 scalar methods, besides the perpetual negative sign before 

 scalar products.) The combination need not be of mere linear 

 circuits, in which differences of current- density are insen- 

 sible ; there may, for example, be induction of currents in 

 a mass of metal not connected conductively with a and b, or 

 the same mass may be in connection ; hut in any case it is 

 necessary that the arrangement should terminate in fine wires 

 at a and b, in order that the two quantities V and C may 

 suffice to specify, by their product, the energy-current at the 

 terminals. Even in this we completely ignore the dielectric 

 currents and also the displacement, in the neighbourhood of 

 the terminals, i. e. we assume c = 0, to stop displacement. 

 This is, of course, what is always done, unless specially allowed 

 for. 



Now supposing the structure of the combination to be given, 

 we can always, by writing out the equations of its different 

 parts, arrive at the characteristic equation connecting the 

 terminal V and 0. For instance, 



V = Z0, (98) 



where Z is a function of d / dt. In the simplest case Z is a 

 mere resistance. A common form of this equation is 



/.V +/iV +/ 2 V + . . . =<? C + 9l C +gfi + ..., 



where the/'s and g's are constants. But there is no restriction 

 to such simple forms. All that is necessary is that the equa- 



