12 . Proceedings of the Ro_i/i(l Irish Academy. 



be obtained by subtracting from the ordinary Bessel integral equation the 

 proper multiple of the equation 



XJn{\r) J,Fp{X)(d/db)PX„(- iXh)^ J„(Xp)pf(p)dp 



2-Fp (A) {d/db)Pj„ [Xb) = ^* 



Lt. 



which it is not difficult to establish by the methods of the paper. 



And as, in the latter method just referred to, expansions (52), (53b) of 

 Part I. are required for + n only, a sufficient condition to be satisfied by ^ 

 in the neighbourhood of p = 0, for the validity of (52), p. 243, is that 



|pi^(/3) I dp and I 1 p'">(p) | dp 



J 



should both converge : it must not be forgotten that + n itself may be negative. 

 (Tlie condition stated on p. 243 was intended to require that the above and 



Ijo'Xp) \dp 



also should converge, but it does not fulfil the intention.) 



pp. 243 et scq. In Arts. 5, 6, it is supposed that a is not zero. 

 p. 244. From the coefficient of <p in (58) delete " 1 ". 



Art. 2. Defects of the former Expansions, especially in regard to Differentiability 



and Uniqiieness. 



I proceed to consider more serious defects. 



In the ease of the new expansions obtained (equations (12), (69}, Part I., 

 and (25), (49), Part II.), I suggested (pp. 226, 238) that they are not unique. 



It was also seen (Part I., Arts. 15, 17 ; Part II., Art. 5) that, as in the well- 

 known cases of the simple Fourier expansions, those equations expressing the 

 expansion of an arbitrary function of :c cannot usually be difi'ereutiated, term 

 by term (once or repeatedly';, with respect to x. 



I did not at first give much attention to these points iu the case of the 

 most general results, and, in the preNious paper, merely justified the applica- 

 tions to problems in \ibratory motion of the expansions appropriate to the 

 boimdary condition expressed by the equation 



{djdcc - h)^ = 0, 

 where A is a constant given arbitrarily, and ha^dng in general different values 

 at the two boundaries, if there are two. Indeed, to some extent I regarded 

 the expansions of a more general type than these as curiosities of Pure 



* For particular cases of tliis equation, compare Carslaw, " Fourier Series and Integi-als," 

 j§ 162-164. 



