SCATTERING OF PLANE ELECTRIC WAVES BY SPHERES. 187 



the wave-length. Then we can simplify the general formulae by observing that (3 '5) 

 may be replaced by the approximation 



if I//ST, (l/*r) 2 , &c., are neglected. Further, in the final formulae for the forces, 

 U and V occur only in the two combinations 



M = i 



r 3r r 



where terms of the relative order l/nr have been rejected. 



On substituting from (4'5) we find, to the same degree of accuracy, 



TI.- 1 3U n IK TT e ""' ~ i i \ 2n+l 



M = + - - = U = sin cos <f> - 2 (- 1)" 7 -r A n P' n (cos 0) 



r/H/y /vt .-/y ' ^i / rtfl I 1 \ 



I// / /Xft=:l /( 



and 



Nl 3V,, , "c -TT n , e """ ^, / 1 \ ^//,-t-i /^ Tv / 

 = T~^ = +" V u = sm e sln ~ - 2 ( - !) -7 7T C P n (cos 



7* O7* 7* /c7* rt = l . i -J- i i 



(4-8) 



Then, substituting in the general formulas (2'l) and (2'2), we find that (to our 

 present order of approximation) the radial components of force are zero, and that the 

 transverse components are given by 



v 3M 1 3N 



V = 4. f'~ J 



30 sin 3 

 (4-9) 



J_ 3_M 3N = _ 



sin 30 " 30 



Accordingly the electric and magnetic forces in the scattered waves are at right 

 angles to each other and to the radius, and their magnitudes are related in the same 

 manner as in a plane wave. 



This conclusion might very well have been anticipated ; and for the case of small 

 obstacles of any shape (with constants K, M- differing but little from unity) the 

 conclusion is contained in a paper by Lord RAYLEIGH.* But I cannot find that it has 

 been noticed for the case of spheres of any size, and of any electrical and magnetic 

 constants. 



This may serve to indicate one advantage of the formulae in spherical polars over 

 those in Cartesian co-ordinates. 



The formulas (4'8), (4'9), with the values of A n , C n given by (471), were those 

 used by Messrs. PROUDMAN, DOODSON, and KENNEDY in their numerical calculations 

 quoted in the introduction to this paper. 



* ' Scientific Papers,' vol. 1, pp. 522-536. For a small perfectly conducting sphere the same conclusion 

 is given by Sir J. J, THOMSON, ' Recent Researches,' p. 448. 



