304-305] PERIODIC MOTEON WITH A SPHERICAL BOUNDARY. 555 



When the wave- velocity c is great compared with 2200 c.s., we may 

 neglect <r 2 /V 2 in comparison with 6. The result is the sama as if we were to 

 put /=0, so that the modulus of decay has, for sufficiently long waves, the 

 value l/2i> 2 found in Art. 301. The same statement would apply to 

 sufficiently minute crispations ; but 6 then ceases to be small, and the 

 approximations break down ab initio. The motion, in fact, tends to become 

 aperiodic. 



305. Problems of periodic motion in two dimensions, with a 

 circular boundary, can be treated with the help of Bessel's Func- 

 tions*. The theory of the Bessel's Function, whether of the first 

 or second kind, with a complex argument, involves however some 

 points of great delicacy, which have been discussed in several 

 papers by Stokes f. To avoid entering on these, we pass on to 

 the case of a spherical boundary ; this includes various problems 

 of greater interest which can be investigated with much less 

 difficulty, since the functions involved (the -fy n and W n of Art. 267) 

 admit of being expressed in finite forms. 



It is convenient, with a view to treating all such questions on 

 a uniform plan, to give, first, the general solution of the system of 

 equations : 



(V 2 + h*)u' = 0, (V 2 + /* 2 X = 0, (V' J + h*) w' = ...... (1), 



du' M du/__ (} 



dx + dy + dz ~ 



in terms of spherical harmonics. This is an extension of the 

 problem considered in Art. 293. We will consider only, in the 

 first instance, cases where u f , v', w' are finite at the origin. 



The solutions fall naturally into two distinct classes. If r 

 denote the radius vector, the typical solution of the First Class is 



d d 



d d 



'at- 



(3), 



* Cf. Stokes, I.e. ante p. 518; Steam, Quart. Journ. Math., t. xvii. (1881); 

 and the last paper cited on p. 558. 



t "On the Discontinuity of Arbitrary Constants which appear in Divergent 

 Developments," Camb. Trans., t. x. (1857), and t. xi. (1868). 



