206 Proceedings of the Royal Irish Academy. 



Theorems in Bessel functions analogous to that of Art. 1 are discussed in 

 Arts. 13, 14. 



These statements, taken together with the Table of Contents below, will 

 probably suffice to indicate the subject-matter of this paper. 



I believe that the method by which eacli integral equation is obtained is 

 valid for functions which are integrable and otherwise satisfy Dirichlet's 

 conditions, and not for any others : I must admit that I cannot speak 

 confidently on these points. 



The method is applicable to developments as integrals, and hence as series, 

 in terms of Legendre's functions, and apparently admits of other extensions* : 

 I hope to return to the subject on future occasions. 



Contents. 



1. Arbitrary function, («), for positive x, expressed as integral whose 

 element is a multiple of {6'(X) cosAa; + S{X) ^m.\x]dX, C, >S' being given 

 polynomials, subject to certain conditions. 



2. Values of the above integrals when o: is negative. 



3-6. A new investigation of the Fourier-Bessel integral theorem. 



3. The equationf : — 



Lt r^ f tt' 



;. = x' \Kn (- iXr) dX K,, {i\p) p(p (p) dfj, = - f {r - e), 



cj-h J a -i 



71 unrestricted, r > a. • ■ - ' - 



4. The equation : — ■ 



Lt. 



XKn (- iXr) dX K'n (- iXp) p^ (p) dp = 0. 



-A 



5. The equation : — 

 \J„ ( A r) dX Jn (Xp) p (p{p)dp^^{l - e'"-') ^ (r - t) . 



6. Equations for a range of from r to h corresponding to those of 

 Arts. 3-5. 



7. Forms assumed by preceding equations when n is an integer. 



8. An alternative discussion : Sommerfeld's investigation extended to am 

 unrestricted n. 



9. Extensions of equations of Arts. 3-6, the A''s being replaced by 

 differential coefficients of any orders. 



* As to the expansions required in discussing the vibrations of elastic solids of certain forms, 

 t 1 distinguish, M'here necessary, between line and contour integrals by ijrefixing the suflSx l or c 

 to the sij^n of integration. 



