193] CASE OF A POLAR SEA. 321 



Assuming S n = (I - prf 8 w COS \ s<* ......................... (viii), 



sin j 



the differential equation in w, which is given in Art. 87, becomes, in terms of 

 the new variable, 



=0 ...... (ix). 



The solution of this which is finite for 2 = is given by the ascending series 



( (n-s)(n+s+I) (n-8-l)(n 

 4 V" 17(+1)~ 1. 



= AF(s-n, s + n+I, s + 1, z] ................................................... (x). 



Hence the expression adapted to our case is 



s . n g<B 



1.2. (* + !)(*+ 2) JsmJ 



where the admissible values of n are to be determined from the condition 

 that d/dd=0 for 6 = 6 V 



The actual calculation of the roots of the equation in n, for any arbitrary 

 value of lt would be difficult. The main interest of the investigation consists, 

 in fact, in the transition to the plane problem of Art. 187, and in the connection 

 which we can thus trace between Bessel's Functions and Spherical Harmonics. 

 If we put a = 00 , ad = r, we get the case of a plane sheet of water, referred to 

 polar coordinates r, . Making, in addition, n6 = kr, so that n is now infinite, 

 the formula (xi) gives 



C 



lcos| 

 "*J sin J 



2(25 + 2) 2.4(2s + 2)(2s + 4) 

 or fxJg(M C ^ 8 \sa) (xii), 



in the notation of Art. 187 (4). We thus obtain Bessel's Functions as limiting 

 forms of Spherical Harmonics of infinite order t. 



* When n (as well as s) is integral, the series terminates, and the expression 

 differs only by a numerical factor from the tesseral harmonic denoted by 



TU (M) . c su, in Art. 87. In the case s=0 we obtain one of the expansions of the 

 sin 1 



zonal harmonic given by Murphy, Elementary Principles of the Theories of Electri- 

 city..., Cambridge, 1833, p. 7. (The investigation is reproduced by Thomson and 

 Tait, Art. 782.) 



t This connection appears to have been first explicitly noticed by Mehler, 

 "Ueber die Vertheihmg der statischen Elektricitat in einem von zwei Kugelkalotten 

 begrenzten Korper," Crelle, t. Ixviii. (1868). It was investigated independently by 

 Lord Eayleigh, " On the Relation between the Functions of Laplace and Bessel," 

 Proc. Lond. Math. Soc., t. ix., p. 61 (1878) ; see also the same author's Theory of 

 Sound, Arts. 336, 338. 



L 21 



