the Magnetic Influence on Spectra. 509 



dppdp P + dz 2 ~° dt 2 ' 



which is ( v 2 -p~ 2 ) H = c 2 d 2 H/dt 2 , 



where v 2 i s Laplace's operator. But a more convenient 

 reduction comes on substituting H = dY/dp, and then 

 neglecting an irrelevant operator d/dp along the equation : 

 this gives 



V 2 Y = c 2 d 2 Y/^ 2 . 



We can now express the disturbance emitted by an electric 

 doublet situated along the axis of z at the origin, and vibrating 

 so that its moment M is an arbitrary function of the time. 

 As regards places at a finite distance, the doublet may be 

 treated as a linear current-element of strength dM./dt. 

 Close up to such an element in its equatorial plane, the 

 magnetic force H due to it is —r~ 2 dM/dt. The appropriate 

 solution for Y for this simplest case is Y =r~ f{t— r/c), 



so that 



(_ r- cr 



giving when is \ir and r is very small, H=— r~ 2 f(t) : 

 thus dM/dt=f(t). That is, if the moment of the oscillating 

 doublet is given in the form dM/dt=f(t), the magnetic force 

 thus originated at the point (r, 6) is 



H=-sin0 



ff(t-r/c) . f(t-r/c) \ 

 \ r 2 + " cr J ' 



or sm0 -r-^t-r/c). 



ar 



The second term is negligible for movements of slow period, 

 as it involves the velocity c of radiation in the denominator. 

 The components of the magnetic field due to a vibrating doublet 

 M at the origin whose direction-cosines are {I, m, n) are then 



(mz — ny, nse—lz, l^—mx)r- 1 d/drr- 1 f(t — r/c), where dM/dt=f(t) ; 



and the components of the magnetic field, and therefore of 

 the radiation emanating from any system of electric oscilla- 

 tors vibrating in any given manner can thence be expressed 

 in a general form by integration. At present we only want 

 the effect of suddenly establishing the doublet M=ev8t at 

 the origin. This comes by integration over the very small 

 time of establishment ; there is a thin spherical shell of 

 magnetic force propagated out with velocity c, the total 

 force integrated across the shell being exactly — Mr -2 sin 6 

 whatever be its radius, for the integral of the second term 

 Phil. Mag. S. 5. Vol. 44. No. 271. Dec. 1897. 2 P 



