212 I^ew Mode of Expressing Solutions of Lwplace' s Equation, 



If in this we put/m(e) = log ^ we obtain harmonics of zero 

 degree. 



These harmonics are given by the series 



r=:x m-\-'2r w-j-r— 1 



(-1) (m + 2r-l)! 



•^•»(4)'<'«-„!,© 



V*^^/ r\ [m +?') I 



r=oo m + 2r m + r-1 



= 2 /tan e\ (-1) Cm + 2r~]j!^ , (5g), 



r=0 \ 2 / /•! [ni + r)] 



except when m = 0, when the first term = log 2". 



Another interesting sequence of harmonics is that given by 

 putting 



/„(--) = -'log,-. 



If 71 be integral, these may be expanded in a series homo- 

 geneous in p and z, the coefficients being harmonics of zero 

 degree, except the last {n — m), which involve log r. Thus : 



J-(p-^|7yiog^ 



=^-J.(p£)log.c 



+ npz-'jj(^pj^y ogz 



^-4#-.V-J./^(4)log. 



+ 



?i(n — l)...{m + l) „ T ., ^/ d\-. 



(n — m)l ' y dz/ ^ 



+ ■ . . 



+ p^^J.^«{p^Jlog.r (59) 



and using the reduction formulae for Bessel coefficients, the 

 quantities multiplying, the various powers of p may be ex- 

 hibited as the sum of terms of the form 



d 



Jr(p^^;^jlogr. 



Such functions will be suitable in problems where the plane 

 ^ = is a locus of discontinuity. 



21. Although the greater number of the results obtained 



