534 Dr. J. W. Nicholson on the Bending of 



For the oscillator alone, 



7 = 



'dp K 



= -*ife^ g sin&- I * E , .... (55) 



taking the important term. 



The intensity at any point, and consequently the mean 

 energy per unit volume, is therefore quadrupled in this 

 region, following the inverse square law from the centre of 

 the sphere. 



This result was to be expected, for in calculating the first 

 approximation, the sphere may be regarded as an infinite 

 plane. The effect of a plane (perfectly opaque) in quad- 

 rupling the intensity is well known. 



But although it is only a first approximation, the formula 

 is very accurate even for a large orientation from the axis of 

 the oscillator. For example, in the numerical case typical 

 of Marconi's experiments, the error is of relative order 10~ 12 

 for all points possessing this type of solution. In such a 

 case, the necessity for a second approximation disappears. 

 The result is very useful also as a justification of the mode of 

 summation of the harmonic series, and as a very compre- 

 hensive check on the accuracy of the work. In a later paper, 

 the mode of approximation to higher orders will be given. 



Points near the axis. 



For points in the neighbourhood of the axis through the 

 oscillator and its antipodes, is small, and it has already 

 been shown that the formula (39) must be employed. Points 

 very close to the oscillator are again excluded. Thus if 



g[6)(D —lca^r ^J VW(cos<£-cos0)- * < 56 > 

 then 



yp = g{0)T i u(e isv * + e igv * + #»* + e izv <) 



Where «=>(BW (57) 



K V 1> V 2) = 4>n — <l>nr±™>$ 



K^3 ? « 4 ) = <t>n—<i>nr+2Xn± m< l>> 



in which ^ n may be neglected as before, in the neighbourhood 

 of a vanishing derivate of an exponent. 



The only series which can have such a vanishing derivate 



