228 Proceediugs of the Royal Irish Academy. 



A discontinuity or boimdaiy value in 0(^')(r) at /', gives rise to a part of 

 the coefficient in the development of 0(7-) whose dominant term, when A is 

 large, is a multiple of 



cos cos //;\//r\/ 



(73) 



Here, again, it is legitimate to differentiate successively under the sign of 

 integration, so long as the usual conditions for the similar process with the 

 Fourier integral are satisfied. 



Aet. 18. RcmarJcs on Discontinuifies in Physical ProUems. \ 



In some physical problems, the condition of Arts. 15, 17, that (p should 

 vanish when x is zero, (sufficient for differentiation), is not satisfied, and ^ 

 may, moreover, be subject to discontinuity in the interior of the medium. 

 It is therefore a matter of some interest to justify the treatment alluded to 

 in the Introduction; any such discussion must, moreover, have a distinct 

 bearing on the question of the validity of Fourier processes generally. 



As regards the effect of discontinuities in value in the body of the medium, 

 it appears perfectly legitimate to state that in all such problems they may be 

 left to look after themselves ; and that a solution, otherwise valid, cannot fail 

 on account of discontinuities, that is, provided they are such as it has been 

 agreed to consider permissible physically. All such discontinuities are to be 

 regarded as a limiting case of rapid variations ; any objection to this line of 

 argument appears to cut at the very root of any mathematical discussion of 

 problems in vibratory motion, (or any other branch of molar physics), which 

 involve discontinuities. For it seems that only by such a procedure can we 

 arrive at the conclusion that a surface of discontinuity is propagated at all, 

 and, as is further necessary, deduce the wave-velocity and the conditions at 

 the surface. Against this view, it might be urged that Love,* following 

 Christoftel,t has discussed this question satisfactorily on the assumption of 

 an absolutely sharp discontinuity, and that he uses only the equations of 

 impulsive motion applied to an exceedingly thin layer including it. To this 

 I should reply that the work rests on a distinct supposition that the discon- 

 tinuity is being propagated through the thin layer, and that no consideration 

 whatever of external forces solely can prove this or throw any light on what 



* "Mathematical Theory of Elasticity" ; also, ""Wave Motions with Discontinuities at "Wave 

 Fronts," § 16, Proc. Lond. Math. Soc, ser. 2, vol. i. 



Here and elsewhere Professor Love has done much to elucidate the propagation, as distinct from 

 the maintenance, of disturbances. My remarks are not to be interpreted as an objection to his 

 procedui-e. 



t " Aimali di Mathematica," 1877. 



