Theory of a Contractile ^Ether to Optical Problems. 531 



equivalent to an increase in its rigidity. Thus, suppose we 

 have a magnetized steel spring vibrating, by placing it in 

 water we increase the effective inertia of the spring, but by 

 placing it in a magnetic field we may stiffen the spring. To 

 express the same in analytical terms the solution of our 

 differential equation is to be, supposing we have a wave tra- 

 velling parallel to the axis of z, 



u = ksmn(z—Yt). 

 And this is a solution of 

 d* C 7 d 2 d i , 1 



p being the density of the aether in free space, B its rigidity, 

 and a, b, a', V, &c. constants. These terms or some of them may 

 give the action of the matter on the aether, those in a, b, c 

 &c. enter the equations as an effective increase of density, 

 those in a', b', &c. as an increase of rigidity. 



To state the same fact in another way, the equations of 

 motion of the aether may be written 



P J=(A-B)g+BV% + X, . . . (39) 



where X represents the action of the matter on the aether. 

 X is to have such a form as will allow the propagation of 

 waves without the absorption of energy and with a velocity 

 independent of the amplitude. It must also give us the 

 ordinary laws of reflexion and refraction, and we must be able 

 to explain by simple hypotheses the laws of double refraction, 

 dispersion, anomalous dispersion, and metallic reflexion. 

 Of late years a number of attempts have been made to find, 

 an expression for this quantity X. An account of them is 

 given in my Report on Optical Theories (B. A. Report 1885). 

 The most complete in some respects is that of Voigt, who, 

 starting with the question as to what is the most general 

 form consistent with the conditions imposed by the problem, 

 comes to the conclusion that for an isotropic medium we 

 may put 



rfV-U) rf»( M -.U) , d\u~\j) TJN 



X=-r— sr -+a-^ ? — +a'-A- w __ n(w _U;, (40) 



where U is the displacement of the matter particles in the 

 same element of volume as the aether, which has u, v, w for 

 its component displacements. In a crystal other terms 

 come in and the coefficients of these may be functions of the 



