24 Proceedings of the Royal Irish Acadeiiuj. 



terms which permit differentiation twice under the integral sign. At excep- 

 tional instants the differential equation has to be replaced by the condition 

 which holds at a wave-front, i.e. that one of the two expressions 



cd(j>/dx ± d(p/dt 

 is continuous; but, as the necessity for this condition can he established only 

 by regarding discontinuity as a limiting case of rapid variation, too much 

 importance might, in my judgment, be attached to its verification. 



If there is only one particle at each end, ^ gives the complete solution. 

 I have verified that, in the case considered by Lord Eayleigh, it agrees with 

 his. 



Art. 9. Application to « Prohleni in Heat-Conduction. 



A problem which admits of a solution closely following that just obtained 

 is that of the flow of heat in a system consisting of a conducting bar, and at 

 each end a number of masses, each of one temperature throughout, and 

 exchanging heat with one another according to the law that the rate of 

 heat transference between any two is proportional to the difference in their 

 temperatures, as well as radiating according to the usual law. 



The modifications which have to be made in the expressions (16), (17), (18), 

 in order to obtain the temperatures of any point of the rod and of typical 

 particles at each end, seem so obvious that, in view of the length of the 

 expressions, it appears unnecessary to give them. 



There is, however, one point of difficulty to which I may allude. It may 

 be held that there is here no physical reason for requiring the origuial 

 temperature distribution, i//, to be continuous ; and, if we permit internal 

 discontiauity, there is some difficulty in showing that the differential 

 equation, which now assumes the form 



d^ldt = c-d-^ldx\ (25) 



and the boundary equations which now replace (7) et seq., are satisfied 

 initially* If the time factor is now taken to be e'^'"''', the characteristic 

 values of (u^ when large, have negative real parts which increase indefi- 

 nitely ; and so, for all positive values of t, all integrals which enter into the 

 solution are so highly convergent, that they may be differentiated under the 

 integral sign as often as is desired ; but this does not apply initially. 



One method of surmounting this difficulty is as follows : — We may add 

 to ^, as given by the equation which now replaces (16), terms in respect of 

 each discontinuity in ip which wUl permit the sum to be differentiated under 

 the integral sign, twice with respect to x, or once with respect to t. 



* 'Diis point appears to be frequently overlooked. 



