Orr — Extensions of Fourier'' s and the Bessel- Fourier Theorems. 23 

 If, however, we add to <i>, in respect of each such discoutinnity .r,, the terms 



(47ri)-' e'""dfi 



c 



[ci-{^-")Fa{nf', - ix)- rM!^-»)i^„(^c, ,1)] 



- Ftinc, fi) e-i^i"^-'') I /jir'4^"(xi) + c-'fi-^x'(^^) I ] ^ (Denominator of (16) j 



. (24) 



in which the vertical hars denote the increase which the expression between 

 them receives as x increases through the discontinuity, and make corresponding 

 additions to Y^, etc., the sum of <p and these terms can be differentiated twice 

 everywhere with respect to t under the integral sign ; and the same is true of 

 these terms themselves at x = a, since, when 



e>^(--<')Fa(fxc, -,i)- e-'^^^-"' F„{fxc, fx) 



is replaced by 2fiTAn, the polynomials in the numerator are four dimensions 

 less than those in the numerator. Thus the instants referred to do not 

 constitute real exceptions. 



Similarly for the equations which hold at the end h. It must be borne in 

 mind that the coefficient Tdjdx, which occurs in (7), appears with the opposite 

 sign in the analogous equation at the end h. 



Art. 8. This Sohdion satisfies the Eqiiation of Motion of the String. 



Finally ^ satisfies the differential equation (12) at internal points. For the 

 integral giving (p can be differentiated twice under the integral sign, except 

 at certain instants corresponding to those at which oiiginal discontinuities 

 in i//" or y' I'cach the point considered after any number of reflexions. 

 (And if such discontinuities are to be considered, distinctions must be drawn 

 between progressive and regressive derivates in the equation.) 



It should be noted that at the boundaries the differential equation (12) 

 cannot be insisted on : it holds, however, except at certain instants. The 

 integral obtained by differentiating ^ under the integiul sign twice with 

 respect to t is, of course, the same as that obtained by differentiating twice 

 witli respect to x, but, althougli when x = a, it is uniformly convergent for 

 all values of t, yet, when t is fixed, it is not uniformly convergent for all 

 values of x, and there are instants at which tlie failure occurs in the 

 neighbourhood of a ; one of these is t = Q. 



If we regarded discontinuity as permissible in yp' or y^ at internal points, 

 we could show by a similar artifice to that of the preceding sections that the 

 differential equation (12), and also equations (7) et scq., are satisfied, except at 

 certain instants. In such a case, we add in respect of each such discontinuity 



