28 Proceedings of fhr Royal Irish Academij. 



TVTien n is not an integer, tlie integrands in these equations may be 

 expressed in a form which, perhaps, is more familiar, by the aid of equations 

 of the type 



K„{x)K„{ye-^-K„{(ne^)K„{y) = iir(2smmr)-' \I„{x)I^{y) - R„{x)I„{y)} 

 = i-- (2 sin n-n-)'' ', J„ [ix) Jl„ {iy) - Jl,, (ir) ./„ {iy) \ . 



The solution may be verified by arguments* following as closely as possible 

 those used in connexion with the problem of Article 5. 



In connexion with those steps in the proof which require the integral 

 in p to be integrated by parts it may be convenient to introduce a repeated 

 integral involving Bessel functions, viz., that value of 



//J . . . (d^i)'xK„(x) or 7r(2 sinw-r H! ■ ■ ■ {dxyx(L„(x) - I„(:c)) 

 which, for arguments of x between - 37r/2 and -f 3n-/2 exclusive, is 

 asymptotically of the form (-)'(7ra;/2)*e"^. 



Ap.T. 13. Expansions applicable to Vibrations of Elastic String itself subject 



to Viscous Forces. 



The theorems which have been given in this paper and in the precediug 

 may be extended to give expansions in terms of the functions which are 

 appropriate to the vibrations of systems similar to those already discussed, 

 in cases where the whole system is subject to viscous forces of the usual type, 

 and probably other expansions of less physical interest. 



Suppose that the problem of Article 5 is altered by having equation (12) 



replaced by 



cT-^ldt- = c-d'(i,/dar - fd,p/dt + gd^<j>/d^dt, (34) 



wherein the second term in the right-hand member is due to a resisting force 

 proportional to the absolute velocity, and the third to forces in each cross- 

 section resisting shear and proportional to the rate of shearing ; also by 

 alteiing the last term of the coefficient of F, in (7) to 



- (T (c- + gD) d/dx, 

 where a- denotes the linear density ; of course a-c' equals T, the tension. 

 The type-solution of (34) is e*^*"', where 



'*'' v = i \g^- -f± \(gfi' -fy + 4cy i^j, (35) 



and the characteristic values of fx, v are determined by 



e'^"'-'''F,(v, ,,)Fa{v, -n)- 0^'-''-»'Ft{v, - fx) FJv: /,) (36) 



in conj\inction with (35). 



* I h:iTe not, however, examined the case in which a is zero. 



