OF EVERY DEGREE OF TRANSPARENCY. 213 



If, then, ij, f denote the components of the actual displacement at 

 the point considered, 



will represent the average values of these components in the small 

 sphere about that point. These average values we shall treat as 

 functions of the coordinates of the center of the sphere and of the 

 time, and may call them, for brevity, the average values of , tj t f. 

 But however they may be designated, it is essential to remember that 

 it is a space-average for a certain very small space, and never a time- 

 average, that is intended. 



The object of this paper will be accomplished when we have 

 expressed (explicitly or implicitly) the relations which subsist 

 between the values of [] Av e> MAVB* [f]Ave> a ^ different times and in 

 different parts of the field, in other words, when we have found 

 the conditions which these quantities must satisfy as functions of 

 the time and the coordinates. 



3. Let us suppose that luminous vibrations of any one period * are 

 somewhere excited, and that the disturbance is propagated through 

 the medium. The motions which are excited in any part of the 

 medium, and the forces by which they are kept up, will be expressed 

 by harmonic functions of the time, having the same period,t as may 

 be proved by the single principle of the superposition of motions 

 quite independently of any theory of the constitution of the medium, 

 or of the nature of the motions, as electrical or otherwise. This is 

 equally true of the actual motions, and of the averages which we are 

 to consider. We may therefore set 



* There is no real loss of generality in making the light monochromatic, since in every 

 case it may be divided into parts, which are separately propagated, and each of which 

 is monochromatic to any required degree of approximation. 



t It is of course possible that the expressions for the forces and displacements should 

 have constant terms. But these will disappear, if the displacements are measured from 

 the state of equilibrium about which the system vibrates, and we leave out of account 

 in measuring the forces (and the electrostatic potential) that which would belong to the 

 system in the state of equilibrium. To prevent misapprehension, it should be added 

 that the term electrical displacement is not used in the restricted sense of dielectric 

 displacement or polarization. The variation of the electrical displacement, as the term 

 is used in this paper, constitutes what Maxwell calls the total motion of electricity or 

 true current, and what he divides into two parts, which he distinguishes as the current 

 of conduction and the variation of the electrical displacement. Such a division of 

 the total motion of electricity is not necessary for the purposes of this paper, and 

 the term displacement is used with reference to the total motion of electricity in a 

 manner entirely analogous to that in which the term is ordinarily used in the theory of 

 wave-motion. 



