SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 
193 
force is equal to S, perpendicular to the plane; thus S is represented hy the graph of 
the Cartesian component Y. 
In general, the resultant electric force is represented by 
-(R^ cos^ 0 sin^ (f). 
5. (iii.) Second Approximations for Conducting Spheres. 
The foregoing algebra needs no alteration beyond replacing f'K.hj the appropriate 
complex refractive index associated with the particular metal and wave-length 
considered. This of course assumes that | K | is not so large tliat the convergence of 
Si (/fitt) 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 
cos 6 = f 
1 5 
{K-l){K + 2 ) 
2K + 3 
fa)'^ 
will not usually give a real value for 6 : and so there is usually no direction in which 
the scattered wave is zero* (or of order faf). 
It may be of interest to note here that experimental work on the scattering 
of light by fine particles has been carried out with silver particles suspended in 
water.! The corresponding values of Ka seem to vary from \ to 2, and the value of 
x/K is taken as 0'2 —i -(3'6); thus the a^pproximations in (ii.) are not sufficient to 
calculate either S„ fa) or S„ («-i«) with any accuracy.| In actual fact it proved 
necessary to use Lord Layleigh’s exact formulte, 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 MacdonaldH for the values of the functions S„ {z), E„ {z), when both 7 i and 
2 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 formulse given on that jrage 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. 
+ To obtain an accuracy of 1 per cent, in Si (z) by retaining two terms of the series only, it must be 
supposed that \z\ does not exceed 1'3. 
§ E. T. Paris, Ioc. cit., p. 472. 
li ‘Phil. Trans.,’ vol. 210, A, 1910, p. 134: the formulae are due to L. Lorenz originally. Avery 
interesting method of deriving the results is given by Debye (‘ Math. Annalen,’ Bd. 67, 1909, p. 535). 
2 E 2 
