6 DcWitt B. Brace, 



Loss of energy may also take place in other ways depend- 

 ing on imperfect elasticity alone, or the loss may arise both 

 from viscous forces and from imperfect elasticity. In the 

 one case we have the stress varying with the rate of distor- 

 tion, and in the other, with the duration and magnitude of 

 the strain. The existence of either will give a differential effect 

 for the absorption of different rays. 



Suppose that absorption does take place, the amplitude of 

 a periodic motion would be some function of the distance, 

 wave-length or period, and of the viscosity and imperfect 

 elasticity. Let it be required to find the form of the function 

 for parallel rays of light, propagated in the direction of the 

 /-axis and with a displacement ^ parallel to the x-axis. 



Let 



i = AC'f'F{y,\,i,), (i) 



where Ae~'^' represents a periodic motion at the origin, of 

 amplitude A, and [x the coefficient of viscosity. When dissi- 

 pative forces proportional to the relative velocities are present, 

 the form of the function F is readily obtained. If the ether 

 is perfectly elastic, the ecjuation of motion for parallel rays is 



P — ; = '^ — ) • (2) 



where p is the density and ;/ the rigidity of the ether. Now 

 Stokes has shown in his celebrated paper " On the Friction 

 of Fluids in Motion " ^ that the expressions for the stresses 

 in an isotropic solid may be obtained directly from those 

 found for the case of a viscous fluid in motion by merely 

 substituting the displacements ^, t], ^ for the velocities ?(, v, w, 

 and the rigidity « for the coefficient of viscosity /x. In the 

 case under consideration, we have not only the rigidity w, but 

 a viscous coefficient /x, each of which produces a shearing 

 stress independently, so that the resulting stress will be the 

 sum of the two, and the equation of motion becomes 



1 Collected Papers, Vol. I. 



6 



