232 Proceedings of the Royal Irish Academy. 



from discontinuities in ■)(, and the same argument as before applies; the 

 second differentiations under the integral sign now are invalid at points 

 X = Xi + ct, o: = «2 -' ct, where x^_ is a discontinuity in ^'. 



Thus ^, given by (73), satisfies the differential equation. 



And it satisfies also the boundary and initial conditions, since the values 

 of cl^lclx, clxpldt are obtainable by differentiating under the sign of integration. 



The cognate problem in Bessel functions, relating to a space bounded both 

 externally and internally, will be discussed in Part II. 



Aet. 20. A Connexion hetiveen Fourier^s and Frullani's Integrals. 



The form of Fourier's integral theorem in which, in the notation of the 

 paper, the element of the integral is g'-^l^""), and the limits of A, are ± co , 

 may be regarded as a particular case of somewhat similar theorems in which 

 the limits of \ are any two infinities, whose ratio is finite, and whose argu- 

 ments both lie within the limits 0, tt, inclusive. Or, changing from i\ to ju, 

 the following equations hold : — 



rhb rx 



d,x\ e^ (»-^) (?0 f^^^ = log ( W -(pix-e), (77) 



J 



Lt. 



A = oo 



Lt. 



ha 



Chh 



dfi e» (^ - ») (^ [u) dii = log (h/a) . (p (:c + e), (78) 



ha 



di 



e->'i''^^)<p[u)dii = Q, (79) 



provided the arguments of a, h both lie between the limits ± -/2, inclusive. 

 It is evident that the two former reduce to 



-hb 



Lt. 



h-<^ 



djx 



e-^yF{y)dy = log [Ija) . F{e), (80) 



where c is any positive quantity. As for proof, it is readily seen that if, in 

 the left-hand member of this, F(i/) is replaced by unity, its value simply is 

 log (h/a,). And, by changing the order of integration,* and applying the 

 theorem of mean values, as in the proof of Fourier's theorem, to the real 

 and the imaginary parts separately, the result readily follows. 



And equation (79) obviously follows from the fact that, if two different 

 positive quantities are substituted in (80) as upper limits of i/, the results are 

 identical, and the difference zero. 



These equations give, however, different expressions for the element of 

 the integral, according as u > or < x; unless a, h are wholly imaginary, 

 numerically equal, and of opposite signs, in which case they reduce to 

 Fourier's. If a, h, are real and positive, the integrals become cases of 

 FruUani's, on integrating with respect to /u first. 



* This step is unnecessary, and Fourier's theorem itself may be proved as indicated here. 



