Our — Extensions of Fourier'' s and the Bessel-Fourier Theorems. 245 



where a<r<'b, subject to the boundary conditions clcpldr - /ti^ = for r = a, 

 d(pldr - 7^2^ = for r = h, and with the initial conditions, <l> = ipy (t<^l^'i = X' 

 tlirougliout the medium at i! = 0. I suppose that ip, dxp/dx, \ are continuous, 

 but permit discontinuities, though not infinities, in d^\p/dx- and d\jdx ; I 

 suppose, also, that all these functions satisfy Dirichlet's conditions. A similar 

 problem for the case in which there is no external boundary, but in which the 

 expansion is trigonometrical, is discussed in detail in Part I., Art. 19. 

 Here, the elementary type solutions may be written 



J.i\r)[\J'^,i\h) - hJJXh)] - J.n{\r)\XJ',J.Xh) - A^/^CAS) }] ^^'^'J, (59) 



J sin Act 



where the admissible values of A are determined by 



{\J\.{Xb) - JhJn(Xh)} {XJ'.uiXa) - h,J_n{\a)\ 



- {XJ'_n{Xb) - hJ,n{Xh)\ \XJ\{Xa) - hJ„{Xa) j = 0. (60) 



I consider, first, the terms which arise from xp. Expressing ^ as a sum, 

 by the aid of (49), we have 



2 R h \j„{Xr) { XJ'-„{\b) - hoJ-„iXb) J - / „(Ar) { \J'„{Xb) - hJn{Xb) J j 



\j„{Xp){XJ'_„{Xa) - lHJ„{Xa)] - J_„{Xp)\XJ',„{Xa) - /hJ„{Xa)\[ pf{p)dp 



~ \{XJ'„{Xb) - /hJniXb)} {XJ'_„{Xa) - hJ^iXa) 

 - [XJ'_„{Xb) - hJ^,lXb)] [XJ'„{Xa) - JhJniXa)] 



= 7r ^ 2 sin?*7r . i/'(r). (61) 



This equation holds at r = a, r = b. This follows from the manner in 

 which it was obtained ; the factor involving r, h varies asymptotically as 

 cos A {r - b) ; this is replaced by 



Of this, for values other than b, only one or other term need be considered ; 

 but when r = b, both terms are of equal weight, and thus the integral with 

 respect to p in the range from a to r, which ordinarily equals only half the 

 right-hand member, now amounts to the whole of it ; this compensates for 

 the fact that there is no range from r to b. Similarly at r = a. 



It is not d priori evident that (61) may be differentiated term by term, 

 for the condition shown to be sufficient, i.e. that xp should vanish at the 

 boundaries, is here violated : this, and the same fact for x^ constitute the 

 chief difficulties. 



For the purpose of ascertaining the order of magnitude of the terms 

 in (61) differentiate each twice with respect to r, taking the portions which 

 involve J±n{Xfi) separately. We obtain the residue of a quantity which, 



R.I. A. PROC, VOL. XXVII., SECT. A. [35] 



