440 The Theory of Aether and Electrons in the 



traction. After this Lorentz* went further still, and obtained 

 the transformation in a form which is exact to all orders of the 

 small quantity w/c. In this form we shall now consider it. 

 The fundamental equations of the aether are 



div d = 4irc*p, curl d = - h, 



div h = 0, curl h = d/c 2 + 47r/ov. 



It is desired to find a transformation from the variables 

 x, y, z, t, p, d, h, v, to new variables x lt y^ z ly t h p lt d,, hi, YI, such 

 that the equations in terms of these new variables may take 

 the same form as the original equations, namely : 



divi d x = 47rc 2 jOi, curl, di = - dh^d^, 

 d^ h! = 0, curlj hi = (1/c 2 ) Bdj/9^ 



Evidently one particular class of such transformations is 

 that which corresponds to rotations of the axes of coordinates 

 about the origin : these may be described as the linear homo- 

 geneous transformations of determinant unity which transform 

 the expression (x 2 + if + z z ) into itself. 



These particular transformations are, however, of little 

 interest, since they do not change the variable t. But in place 

 of them consider the more general class formed of all those 

 linear homogeneous transformations of determinant unity in 

 the variables x, y, z, ct, which transform the expression 

 (x- + y* + z - c't") into itself : we shall show that these trans- 

 formations have the property of transforming the differential 

 equations into themselves. 



All transformations of this class may be obtained by the 

 combination and repetition (with interchange of letters) of one 

 of them, in which two of the variables say, y and z are 

 unchanged. The equations of this typical transformation may 



* Proc. Amsterdam Acad. (English ed.), vi, p. 809. Lorentz' work was 

 completed in respect to the formulae which connect pi, vi, with p, v, by Einstein, 

 Ann. d. Phys., xvii (1905), p. 891. It should be added that the transformation 

 in question had been applied to the equation of vibratory motions many years 

 before by Voigt, Gott. Nach. 1887, p. 41. 



