Longitudinal Component in Light. 263 



wave that is thrown off into space by the rotation of each 

 of the earth's magnetic poles. 



The more complex wave thrown out by the earth with its 

 two magnetic poles comes under the next head ; but it is 

 waves of this type which must be thrown off by the planets 

 rotating round the sun, if they are electrified, and by their 

 gravitating property if gravitation be propagated in the same 

 way as electromagnetic disturbances. 



III. We can produce any desired combination of complex 

 doublets by operating on a simple doublet with a function 



of (— j -7-j ~\, The typical term of such a function may 

 be taken as 



If we write 



\dx)\dy)'\dz)~ 

 _ cos (pt — qr) 



r 



we get as a typical case, 



F=Sw, G = H = 0. 



Also, remembering that A <2 u + q 2 .u = 0, we have for the 

 electric force corresponding to this typical case of a vector 

 potential, 



x dor dxdy dxdz 



Now this operation will introduce all sorts of powers of - 



and of q, and I only want to calculate the principal term in 

 the longitudinal component. In making this approximation 

 we may simplify the calculation by observing that the largest 

 terms are always due to differentiations with respect to the 

 circular part of u, and that differentiation with respect to 

 x, y, z, or r lowers a term by one. We may then leave 

 out all differentiation with respect to coordinates outside the 

 circular part in terms of the second order, and it is well to 

 reduce the differentiations represented by 8 so as to produce 



i. e .u, e * cos (pt — qr) , * sin (pt — qr) r . P 



terms of the form o — ±-^ and o — — - J — L . Of 



r r 



course it very much simplifies calculation to use the typical 

 form e~ iqr for the circular functions. 



We thus get for the values of the components of electric 

 force to the second order : — 



