Expansion in BesseV's Functions, 549 



validity of (iiiV) in the limit when r = a, 



£ rf(r)B Q (fi k r)dr^pf(a)B,(fi k a) 



Through lack of this expansion the solutions of certain 

 problems have hitherto been left incomplete. Consider, for 

 example, the symmetrical motion of a uniformly stretched 

 circular membrane loaded in the middle, i. e. of an annulus 

 to the inner circular boundary of which a load symmetrical 

 about the centre is attached. We have with the usual 

 notation 



subject to the conditions 



£=0, when r=b ; 



9 o^ 

 z~m-^^- „ r = a ; 

 Or ' 



z=f{r) „ t=0; 

 and i — „ t =0, 



if the membrane starts from rest. 

 Hence 



00 



z = X A& cos (wO B o(w)> 

 where ^ , . T , . J (^) 



and the /u's are determined by 



3/0*0 



B (j"a) 



m 



fi. 



Therefore 



f 6 c 2 



A* = 2 ^ J^ ^^ . 



The small plane oscillations of a uniform, heavy, flexible, 

 inelastic string hanging freely with a load attached to the 

 end may be investigated similarly. 



It may be noted that the method employed in obtaining 

 the generalized expansion can be applied to expansions in 

 other functions, if the necessity arises in connexion with any 

 physical problem. 

 June 1907. 



