THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



\,\^ * [JSIXTH SERIES.] 



\ U. s pATrMT tf**£l N UAR Y 1922. 



I. Zonal Harmonics of the Second Type. 

 By J. W. Nicholson, F.R.S.* 



THE function Q n (f^) has received but little study in some 

 of its aspects. In fact, nowhere in the literature of 

 the subject has the writer been able to find the value of the 

 integral 



-1 



j: 



which is clearly fundamental. Together with certain other 

 properties of the function, it was found necessary recently 

 to evaluate this integral in connexion with the solution of an 

 integral equation arising in the problem of two parallel con- 

 ducting disks. The capacity of an electrical condenser of 

 this type depends on these properties, and can be expressed 

 without approximation as an integral containing Q functions. 

 A short account of these properties which, though apparently 

 not on record or perhaps unknown, can be obtained in an 

 elementary way, is given in the present note. 



The function Q n {/n), the second solution of Legendre's 

 equation, may be defined by 



Q, l ( /t )=P>„0)logi±^- 2 «,.?._*„ 



1 — fl r = 



when fju is less than unity, where the coefficients a r are 

 numerical. 



* Communicated bv the Author. 



Phil. Mag. S. 6. Vol. 43. No. 253. Jan. 1922. B 



