188 DR. T. J. I'A. BROMWICH ON THE 



5. SPHERES SMALL COMPARED WITH THE WAVE-LENGTH. 



The fundamental assumption is that xa is small enough to justify us in rejecting 

 all but one or two terms in the power-series for S n (KO) and E n (*a). 



It has been usual to assume further that \^a\ is correspondingly small ; but 

 Dr. PROUDMAN has remarked that in the case of a dielectric sphere with a large 

 value of K the second assumption need not follow from the first. It seems worth 

 while therefore to simplify the formulae (47) by expanding in powers of KO, , while 

 retaining the general forms for S n (/<]) ; it will be seen moreover that the resulting 

 formulse form a link between the results for dielectric spheres and those for 

 conductors. 



Remembering that S n (*c) is of order (/ca) n+1 and that E B (/ca) is of order (/ca)~ n , it is 

 easy to see from (4'7) that in general A n and C n are both of order (/ca) 2 " +1 . Thus in 

 the first approximation it will be sufficient to deal only with the coefficients A] and 

 G! ; and for these we need the formulae for Sj(/cia) and S'i (/qa). Now from (3'4) 

 we have 



a / \ 2/ Id \fsu\z\ sin z sinz/. \ 



b, (Z) = Z*\ --- r - - COS2=- -(12 COt 2), 



\ z dz' \ z / z z 



and so 



.I/ / \ /, 1 \ cos z sin 2/3 \ 



b', (2) = sin z( 1 - J +- - = (21+z cot z). 



\ * / 2 2 



Hence 



, 



bi (2) 12 C()t 2 



where now 



F(z) = z'/([-zcotz). 



In the second place, for the functions of *, from (3'4) and (3'6) we find the first 

 approximations 



B 1 (z) = ^, E 1 (z)=l/z. 



Substituting, it will be seen that for n = 1 the first equation in (47) gives 



3A, 1 g ._ o 3A, 



where 



bi (Kid) 

 After a little reduction the last equation gives 



The second equation in (47) gives a similar formula for C,, with yu taking the 



