570 ScJdo mil cli's Theorem in BesseVs Functions. 



Accordingly 



F(R)=/(O)4-Rf 2 /'(Rsm6V0. . (11) 

 Jo 



That f(r) in (1) may be arbitrary from to it is nov 

 evident. By (3) and (6) 



2 C^ 

 f( r ) == ~\ ^{5 + ft 1 cos(rcos^) + 6 s cos(2rco3^)+ . . .} 

 ""Jo 



= &o-f-&iJoW + M (2^)+... , (12 « 



where 



*.= -f F(f>df, 6„--fw*5F(S)«*e., . . (13) 



""Jo ""Jo 



Further, with use of (11) 



&„=/(<>) + -fV.f.f'/Cfrintf).^, . . (14) 



77 J Jo 



^-.f'^.fcosnf.f'/'Cfsin^rf^ , . (15) 



^Jo Jo 



by which the coefficients in (12) are completely expressed 

 when /is given between and it. 



The physical interpretation of Schlomilch's theorem in 

 respect of two-dimensional aerial vibrations is as follows : — 

 Within the cylinder r = ir it is possible by suitable move- 

 ments at the boundary to maintain a symmetrical motion 

 which shall be strictly periodic in period 2ir, and which at 

 times £ = 0, t = 2ir, &c. (when there is no velocity), shall 

 give a condensation which is arbitrary over the whole of the 

 radius. And this motion will maintain itself without ex- 

 ternal aid if outside r = ir the initial condition is chosen in 

 accordance with (6), F(f) for values of f greater than tt 

 being determined by (4). A similar statement applies of 

 course to the vibrations of a stretched membrane, the trans- 

 verse displacement w replacing s in (5). 



Reference may be made to a simple example quoted by 

 Whittaker. Initially let /{/•)=?•, so that from to ir the 

 form of the membrane is conical. Then from (12), (14), 

 (15) 



2 O 



l> = J > h n = ( W eVe11 ) > K = ~ ~2 ( H ° M ) '> 



