332 The Hon. J. W. Strutt on Bessel's Functions. 



perly speaking, satisfy equation (10) in passing through r = 0. 

 The solution of (12) may therefore be written 



<p = SS J„(/cr) cos {n6 + a) , 



the one summation relating to k and the other to n. We have 



now to satisfy the boundary condition, that -~-=0 when /•=!. 



It appears that only such values are admissible for k as make 

 J'„(/c)=0. Reflecting now that initially (j) may have any value 

 within the circle r=l, we infer that any function of r may be 

 expanded from r=0 to r=l in a series of the form 



A,J„(/c,r) + A2J„(/C2r) + ..., 



/Cp /Cg, . . . being roots of J'„(ac) =0. A similar argument would 

 apply if K were a root of !«(«) =0. A rigorous analytical proof 

 of the possibiHty of these expansions would probably be difficult. 

 The values of the coefficient we shall see how to determine pre- 

 sently. 



Let us consider the functions 



</> = e'"' cos n6 JJkt), H 



(j>' = e-''''cosneJn{'c'r)J ^ ^ 



They satisfy Laplace^s equation. When 2' = they become 

 </> = cosw^.J„(/cr), (j>' = cos n6 J„(/cV), respectively, and they vanish 

 when r=x . If S denote the space bounded by ^=0, 2=00 , 

 and the cylinder r=rf we have, by Greenes theorem, 



Over the plane end 2=0, 



(p = cos n6 J„(/cr), -^ = + /c' cos nO 3n{f<^^) '■> 

 over the cylindrical surface, 



whence 



0=e~**cosr/^ J^(«;r), -^ =k'€~'^'^ cos n6 i'„{Kr); 



Uct> ^ ^S= C^S^'rdedr cos^ nO . k'. J,(/cr)J,(AcV) 



+ i I rdOdze -^''+'''^* cos^ nd . «'J„ (/^r) J'„ (/cV) 

 Jo Jo 



= 7r/c'i r^rJ„(A:r)J„(«'r) + -_-, J^(^r) . J„(«V) 



Jo /CH-/C 



