SCATTEEING OF PLANE ELECTRIC WAVES BY SPHERES. 193 



force is equal to S, perpendicular to the plane ; thus S is represented by the graph of 

 the Cartesian component Y. 



In general, the resultant electric force is represented by 



5. (iii.) Second Approximations for Conducting Spheres. 



The foregoing algebra needs no alteration beyond replacing ^/K by the appropriate 

 complex refractive index associated with the particular metal and wave-length 

 considered. This of course assumes that K is not so large that the convergence of 

 Sj (K^O) becomes too slow to justify the approximation made above ; and then the 

 formulae (5'4) to (5'6) provide the solution. It will be noticed that when K is 

 complex, the equation 



, - / v 



" A 2K + 3 



will not usually give a real value for 9 : and so there is usually no direction in which 

 the scattered wave is zero* (or of order (*)') 



It may be of interest to note here that experimental work on the scattering 

 of light by fine particles has been carried out wjth silver particles suspended in 

 water.! The corresponding values of tea seem to vary from \ to 2, and the value of 

 Y/K is taken as 0'2 i (3'6); thus the approximations in (ii.) are not sufficient to 

 calculate either S n (KO) or S n (/qa) with any accuracy 4 In actual fact it proved 

 necessary to use Lord RAYLEIGH'S exact formulae, equivalent to (4'7) above, and to 

 go as far as n = 4 in the series. 



6. CASE OF LARGE PERFECTLY CONDUCTING SPHERES. 



Before proceeding to the final formulae, it will be convenient to state certain results 

 given by MACDONALD|[ for the values of the functions S n (z), E n (z), when both n and 

 z are large. 



* For the case in which K - 1 is small this conclusion is given by G. W. WALKER, ' Quarterly Journal 

 of Mathematics,' vol. 30, 1899, p. 217. The formulae given on that page agree with (5 -6), when K- 1 is 

 small ; but the more general formulae on the preceding page do not agree with (5 6) completely. I have 

 not succeeded in tracing the discrepancy on account of the fact that G. W. WALKER has omitted some of 

 the details of his preliminary calculations. 



t E. T. PARIS, 'Phil. Mag.,' vol. 30 (Ser. 6), 1915, p. 459. 



I To obtain an accuracy of 1 per cent, in Sj (z) by retaining two terms of the series only, it must be 

 supposed that \z\ does not exceed 1'3. 



E. T. PARIS, loc. cit., p. 472. 



|| 'Phil. Trans.,' vol. 210, A, 1910, p. 134: the formulae are due to L. LORENZ originally. A very 

 interesting method of deriving the results is given by DEBYE ('.Math. Annalen,' Bd. 67, 1909, p. 535). 



2 K 2 



