230 Proceedings of the Royal Irish Academy. 



d(j)ldt = x> throughout the medium. (If k is zero or infinity, the difficulties 

 are reduced to a minimum.) 



For many reasons I restrict the discussion to the case in which \p, dip/dx, x, 

 are continuous ; and I suppose that during the time considered, the disturbance 

 never extends beyond some finite distance x = h; I do not suppose that d^xp/dx^ 

 or dx/dx is continuous, but suppose them finite Dirichlet functions. 



Such a problem may arise in various fields. The discussion of the integrals 

 is germane to that of the series which would arise in problems relating to the 

 space between concentric spheres, for instance ; and this constitutes, perhaps, 

 its chief interest. 



Here the elementary type solutions are : 



,, . . ^ X cosAc^ 



(A cos Aa^ + K sm Ax) . , • 

 smXct 



I consider, first, the term which arises from ip. As in the type solution 

 C is of higher dimensions than S, the value of d-pjdx is obtainable from that 

 of i// by differentiation under the integral sign ; (see Art. 15). This holds even 

 at a? = 0, since, when equation (70) is applied to -pi, the integral on the left is 

 there discontinuous, falling from ^//'(f) to zero ; the part of the right arising 

 from the first term of (71) is also discontinuous, falling from zero to - K\p(e); 

 and, from the boundary condition, supposed to hold for the initial \p, these 

 discontinuities compensate. 



As regards d^pjdx^, the coefficient of 



^ , A cos \x + K sin \x ,. 



'^TT -T ;; dX, 



A,- + K- 



in the integral which gives it, (of type (12), except that the limits of A are 

 and CO ), is 



(A cos \u + K sin \u) \p"{u) du (74) 



J 



= - X^p'{e) - A (k cos Xtt - X sin X'li) ■p'{it) du 



J 



= - A^'(£) + KX\P(e) - A- (A cos X^c + K sin Xu)\P(u)du. 



J 



In virtue of the boundary condition, which is supposed to hold for the initial 

 disturbance also, this reduces to the final term alone. Thus, the value of 

 d^xjj/dx'^ is obtainable from ^ by a second differentiation under the sign of 

 integration, (provided we do not, after the first differentiation, discard from 

 dxp/dx the terms involving \p(e) which, save when x = 0, really contribute 

 nothing to d\p/dx). 



Now, the expansion of -tp'X^) i^ uniformly convergent except near places 

 of discontinuity, as ^//" has no infinity ; and the coefficients of dX in it are 



