8. Let us now examine the relation which subsists between the 

 values of [B] Ave and [U] Ave for the same point, that is, between the 

 average electromotive force and the average displacement in a small 

 sphere with its center at the point considered. We have already seen 

 that the forces and the displacements are harmonic functions of the 

 time having a common period. 



A little consideration will show that if the average electromotive 

 force in the sphere is given as a function of the time, the displace- 

 ments in the sphere, both average and actual, must be entirely 

 determined. Especially will this be evident, if we consider that since 

 we have made the radius of the sphere very small in comparison with 

 a wave-length, the average force must have sensibly the same value 

 throughout the sphere (that is, if we vary the position of the center 

 of the sphere for which the average is taken by a distance not greater 

 than the radius, the value of the average will not be sensibly affected), 

 and that the difference of the actual and average force at any point 

 is entirely determined by the motions in the immediate vicinity of 

 that point. If, then, certain oscillatory motions may be kept up in 

 the sphere under the influence of electrostatic and electrodynamic 

 forces due to the motion in the whole field, and if we suppose the 

 motions in and very near that sphere to be unchanged, but the motions 

 in the remoter parts of the field to be altered, only not so as to affect 

 the average resultant of electromotive force in the sphere, the actual 

 resultant of electromotive force will also be unchanged throughout 

 the sphere, and therefore the motions in the sphere will still be such 

 as correspond to the forces. 



Now the average displacement is a harmonic function of the time 

 having a period which we suppose given. It is therefore entirely 

 determined for the whole time the vibrations continue by the values 

 of the six quantities 



MAVO LSJ 



Avej 



JAve 



at any one instant. For the same reason the average electromotive 

 force is entirely determined for the whole time by the values of the 

 six quantities 



for the same instant. The first six quantities will therefore be 

 functions of the second, and the principle of the superposition of 

 motions requires that they shall be homogeneous functions of the 

 first degree. And the second six quantities will be homogeneous 

 functions of the first degree of the first six. The coefficients by which 

 these functions are expressed will depend upon the nature of the 

 medium in the vicinity of the point considered. They will also 



