124 Mr Basset, On a Class of Definite Integrals [Nov. 13, 



The advantages of reducing a double integral to a single 

 integral are obvious ; and it often happens that when the integra- 

 tion with respect to X has been performed, the integration with 

 respect to ^ or <^ can also be effected, and the integral completely 

 evaluated. 



A good many proofs of (4) have been given, one of which will 

 be found on p. 431 of the fifth volume of these Proceedings. 

 Equation (6) may be established in a precisely similar manner by 

 integrating the definite integral 



/.OO 



I X sin X cos u^ (x^ — r^) dx du, 

 Jo 



first with respect to x, and then with respect to u and comparing 

 the results. 



The function Y^^. 



2. In many investigations the second solution of Bessel's 

 equation, which will be denoted by F^, is required. In some cases 

 it is better to employ complex quantities throughout and to discard 

 the imaginary part in the final result, whilst in others it is better 

 to employ a function with a real argument. It is easily shown 

 that the definite integral 



2 r €-"''= dx 



{^)> 



TTJi (X'-I)^- 



satisfies the same equation as Jo{r), as is otherwise obvious from 

 the theory of linear sources of sound — see Rayleigh, Theory of 

 Sound, Vol. II. p. 275. The integral (8) may be written 



r°° 2i r" 



cos (?* cosh 6) d(p sin (r cosh 0) d^ = Yq (r) - iJ^ (r) 



Jo "^ Jo 



2 r , , ,, ,, 2 



by (4), whence 



2 r 



Fo (^') — — \ cos (r cosh d))d!^ (9), 



7'" Jo 



which may be regarded as the definition of Fo; also differentiating 

 with respect to r, and recollecting that Fi (7^) = — Y^ (r) we obtain 



2 f" 

 Y^{7^) = - I cosh (f) sin (r cosh, (fi) dcfi (10). 



This result will be obtained by a more satisfactory process 

 later on. 



