Forced Vibrations of Electromagnetic Systems. 383 



('•H)"={/+^^- ) },H ) . . . (105) 



» ( ^__«fl££i) H (106) 



The equation (106) has for solution 



where A is independent of 9, and is to be found from (105). 



The most practical way of getting the r functions is that 

 followed by Professor Rowland in his paper*, wherein he treats 

 of the waves emitted when the state is sinusoidal with respect 

 to the time. We shall come across the same waves in some 

 problems. 



Let E = V m e —vC£ m (107) 



r ' 



Then the equation of T m is, by insertion of (107) in (105), 



P" + 2gP'= m( ™ 2 +1) P; .... (108) 



and the solution, for practical purposes with complete har- 

 monics, is 



m(m + l) m(m 2 — V){m + 2) 

 2qr 2 . Aq 2 r 2 



m(m 2 -l)(m 2 -2 2 )(m + 3) , ' nm 



2.4.62V 5 + V / 



We shall find the first few useful, thus : — 



P 2 =l-3(^)-> + 3(^)- 2 , V . (110) 



P 3 = l_6(^)- 1 + 15(^)- 2 -15( ? r)- 3 .) 



Now let U = e ?r P, so that U is the r function in Hr. If 

 we change the sign of q in U, producing, say, W, it is the 

 required second solution of (105). Thus 



IW(l-i), W 1=e -"(1 + I) . . (Ill) 



in the very important case of Q 1? when m = l. 



* Phil. Mag. June 1884, " On the Propagation of an Arbitrary Electro- 

 magnetic Disturbance, Spherical Waves of Light, and the Dynamical 

 Theory of Refraction." Prof. J. J. Thomson has also considered spherical 

 waves in a dielectric in his paper " On Electrical Oscillations and Effects 

 produced by the Motion of an Electrified Sphere," Proc. London Math. 

 Soc. vol. xv., April 3, 1884. 



