Closing Years of the Nineteenth Century. 441 



easily be derived by considering that the equation of the 



rectangular hyperbola 



x 2 - (cty = 1 



(in the plane of the variables x, ct) is unaltered when any pair 

 of conjugate diameters are taken as new axes, and a new unit 

 of length is taken proportional to the length of either of these 

 diameters. The equations of transformation are thus found to be 



x = Xi cosh a + cti sinh a, y = y\ y 



t = ti cosh a + (x } /c) sinh a, z = z,, 



where a denotes a constant. The simpler equations previously 

 given by Lorentz* may evidently be derived from these by 

 writing w/c for tanh a, and neglecting powers of w/c above the 

 first. By an obvious extension of the equations given by 

 Lorentz for the electric and magnetic forces, it is seen that the 

 corresponding equations in the present transformation are 



= d Xlt h x = 



d y = d yi cosh a + ch zi sinh a, 

 d z = d zi cosh a - ch v . sinh a, 



h y = h yi cosh a - (l/c)d zi sinh a, 

 h z = h zi cosh a + (l/c)d yi sinh a. 



The connexion between p and p l may be obtained in the 

 following way. It is assumed that if a charge e is attached to 

 a particle which occupies the position (f, 77, J) at the instant t y 

 an equal charge will be attached to the corresponding point 

 (f i, 77 1} \) at the corresponding instant ti in the transformed 

 system ; so that a charge e attached to an adjacent particle 

 (f + A TI + Arj, %+ A) at the instant t will give rise in the 

 derived system to a charge e at the place 



V %1 A . Ofrl A %l 



fi +^Af + ^-AT/4 ^\ 



at the instant 



* Cf. P . 434. 



