Orr — Extensions of Fourier'' s and the Bessel-Fourier Theorems. 229 



happens in the layer. To do so it is necessary to analyze the layer into still 

 thinner ones, (this being frequently done almost unconsciously), and, I believe, 

 the investigations in question ultimately turn on the possibility of replacing 

 the absolutely sharp interface by a layer of more gradual transition. At a 

 surface of absolute discontinuity we can neither, I think, obtain the ordinary 

 differential equation, nor dispense with it. 



Moreover, looking at the question from the mathematical standpoint, if a 

 function is discontinuous for a certain value x^ of the variable x, (distance or 

 time), it is possible, in the range from Xi - i. to Xi + i, where f, e are any 

 finite quantities, however small, to replace it by another so as to make 

 continuity prevail throughout in the function and its derivatives up to any 

 definite given order which may be required. 



In the manipulation of integrals and differential coefficients, however, the 

 effect of discontinuities must not be overlooked. The criticism directed by 

 Love* against Poisson's and Stokes' discussions of the propagation of an 

 arbitrary disturbance on account of supposed failure in the case of discon- 

 tinuity is based on such an oversight: this has been pointed out by Lord 

 Eayleigh.f 



It does not appear, however, that the discontinuities which arise by reason 

 of definite boundary conditions can be explained away in this fashion. More- 

 over, the direct consideration of discontinuities in the body of the medium 

 would afford a certain amount of verification and be of interest. But, in all 

 cases, questions as to the convergence of integrals would be raised, and, in 

 addition, if the functions to be employed in the expansion are not trigono- 

 metrical, (but Bessel, for instance), some analytical difficulties may be looked 

 for. 



Art. 19. Validity of Discussion of Vilratory Motion hy Integrals of Fourier 

 Type established in a Simijle Case. 



A discussion of the validity of the application of trigonometrical functions 

 to the general problem of the space outside a sphere by the aid of equation (12), 

 as suggested in the Introduction, would be a matter of some complexity ; and 

 the solution by Love's general functions is, in fact, more convenient. I may 

 consider, however, the following simple problem : — A solution is wanted of the 

 equation c-d'^(f>/dx' = d'^<pldf for positive values of x, subject to the boundary 

 condition d(p/dx - K<p = for x = 0, and with the initial conditions (p = '>p, 



* "Wave Motions witli Discontinuities at Wave Fronts"; also, "The Propagation of Wave 

 Motion in an Isotropic Elastic Solid Medium," Proc. Lond. Math. Soc, ser. 2, vol. i. 



t "Note on the Application of Poisson's Formula to Discontinuous Disturbances," Proc. Lond. 

 Math. Soc, ser. 2, vol. ii. 



R, I. A. PROC, VOL. XXVII., SECT. A, [33] 



