an Arbitrary Electro-magnetic Disturbance. 431 



This same series occurs in the expression for the light from a 

 circular orifice; but I am not aware that writers on physical 

 optics have recognized this connexion with Bessel's functions. 

 Prof. Stokes has given the value of the series for large values 

 of br sin 0, and the same value is given by writers on BesseFs 

 functions. We thus find, writing v = brs\n 0, 



4wR V Try € LI 4Jl + 2.4 (4i>) 2 



3.5.7.9.1.3.5 1 , x \ 

 2.4.6.8 ~ (4t>)* J 



. / *r\f3 1 3.5.7.1.3 1 . \1 



The energy given out per unit of time is, by the first formula, 



~2K — I 5 + 35 ~ 720 + 31680 1310400 J' 



and by the second, for very large values of br, 



Yrb 3 Gl 

 2ttK ' 



Let us now reduce the series to spherical harmonics. 

 Write 



.. 27TlV 



u = cr= — ior=z - — , 



A. 



P=m2 {I + 2tV^^ + (2^476 Sin ^ + &C -}' 



This satisfies the differential equation 



<?P lc/P . 



-r-* 7 Psiir = 0. 



du u clu 



Write also /x=cos0 and Q' w = -? — . P can be developed in 

 the following series, as it only contains the even powers of /x: 



P = A' 1 + A / 3Q' 3 + A' 4 Q' 4 + &c. 

 But 



Q - sin2 6 = - (2n-l)(2ltlj(^+3) K^-l)^« 



~(4n+2)Q / n + (2n + 3)Q / w _ 2 }. 



TXT x- * ^ 1 ^ 1. 



Writing 8= — -, r-, we have 



* dvr w aw 



A, '=H{t A '.- sA ''}> 



2G2 



