Ok It — Extensions of Fourier's and the Bessel-Fourier Theorems. 27 



Art. 12. The corresponding Modification of the Expansions in Bessel F^mctions, 

 and application to a Physical Problem. 



Proceeding next to similar modifications of the Bessel sum and integral 

 theorems of the former paper, it will, I think, suffice to consider the problem 

 in Physical Mathematics analogous to that discussed in Articles 5-8. 



Let, then, the differential equation (12) be replaced by 



d-'(l,/dr' + r-'df/dr - nh--^ = c-'d'^/dt\ (30) 



the other conditions to be satisfied being expressed as in Article 5, except 

 that r now replaces x. This would apply to the vibrations of a circular or 

 annular elastic lamina whose boundaries are not fixed, but connected by 

 elastic membranes devoid of mass to a series of concentric rings which offer 

 no resistance to being bent out of their planes.* 



The solution is comprised in the typical equations 



^ = - (2^7 



ci^^dfi 



I 



Faiiiic, d/da) {K„{na)Kn{fJ.r6-') - Kiixae-^) Kn{nr)] 

 Fi(fic,d/db) I K„{nb)K„{fipe-') - K„{nbe-i)K„{fxp) | ( ,x^{p) + c-'x(p) \pdf>+i^bn, 

 - iTT aO, Fii^^c, d/db)[K„(^b)K„ifire-') - K„{ube-')K„{iJ,r)\ 

 4- Fa {uc, d/da)Kn (fia)Fi Qxc, djdb) K,, (/xbe'") 



F„(„c, d/da)K„(fiae'^')Fi{iic, dldb)K„(iJ.b) 



(31) 



„F, = (27r)' U^^'rf, 



'TA, 



iFi,{fic,dldb)\K„{ub)K'„{fj.pe^'')-K„{nbp')K„{fip\\ \nUp)^c 'xip)\p(^p-^i>^> 

 + ian,Fi{fic, djdb) {K„{,.ib)Kn{fire''') - Kn{p.be"')K„{^ir)] ,.J 



Denominator of (31) 



(32) 



5 F, ^ - (27r)-' 



c^^'dfi 



\b-'TB,, 



iF„(nc, d/da) { K„{fia) K„{fipc^'yKJ,.iae^')K„(np) ] { ii^P(p) + c' x(p) \pdp + Tr„U,\ 



+ iiniFa{ixc,dlda){K„(jia)K„{0-e ') - K„{nae^')K„{nr)],.J: 



^ Denominator of (31). (33) 



* I'he equation which now replaces (7) would present itself more naturally in a form in which T 

 is replaced by rT. 



[4*] 



