Orii — Extensions of Fourier'' s and the Bessel-Fouricr Theorems. 207 



10. The cases in which a is zero and h inlinite. 



11. Nature of the convergence of the Fourier-Bessel Integral. 



12. Differentiation of the Fourier-Bessel equation under sign of integration. 



13. Simple example of Bessel Expansion analogous to that in Art. 1 : 



Expression of ^ (r) as Integral wliose Element, as a function of r, is a, 



multiple of 



[Jn (Ar) J_n (ka) - /_„ (Xr) /„ (Aa) } d\, r > a. 



14. A Generalization of the preceding Kesult. 



15. Differentiation of equations of Art. 1 under Integral sign. 



16. jSTature of the Convergence of the Integrals in Art. 1. 



17. Differentiation of equations of Arts. 13, 14 under Sign of Integration. 

 Nature of the Convergence. 



18. Eemarks on Discontinuities in Physical Problems. 



19. Validity of Discussion of Vibratory Motion by Integrals of Fourier 

 type established in a simple case. 



20. A connexion between Fourier's and Frullani's Integrals. 



Art. 1. Arhitrary function, <i> [x), for positive x, expressed as integral ivhose 

 element is a multiple of { C(X) cos Aa; + >S'(A) sin Xx\clX, 0, S being 

 given polynomials, subject to certain conditions. 



In connexion with the above class of Problems in Mathematical Physics 

 the following question suggested itself, viz. : — 



Can an arbitrary function, </)(a;), be expressed, for positive values of r, as 

 an integral in which each element is of the form 



(6' cos Xx + S sin \x) dX, (1) 



where the ratio of C to S is a given rational fractional function of A ? 

 Any such ratio may be expressed in the form 



C/S = 6'(A)AS'(A), (2) 



where 0(X), >S'(A) are given polynomials which have no common zero. 



I suppose that j^ (p{x)dx is convergent, and that 0(^) otherwise 

 satisfies Dirichlet's conditions. 



Consider the integral 



cJ-A ^(a; + lo{A) Jo 



where the path of A is a contour lying on the upper side of the axis of real 

 quantities, and everywhere at a great distance from the origin. 



