due to a Travelling Disturbance. 543 



The frictional coefficient has been finally put =0, but it 

 may sometimes be necessary to retain it, in order to secure 

 the determinateness of the integrals. This necessity will 

 arise whenever U— -ccosi/r vanishes within the range of yfr, 

 as for instance in the case of water-waves subject to gravity 

 .and capillarity combined, if c exceeds the mininuni wave- 

 velocity. 



It is to be remarked, moreover, that we have so far 

 imagined the disturbing influence to be concentrated at a 

 point, with the result that the integrals with respect to m 

 which we have discarded become in some applications 

 infinite when x cos yfr + y sin -*{r=z0. 



This may be remedied by the introduction of a factor e~ kb 

 in (13) and subsequent formulae. The effect of this is that 

 the distribution of the disturbing force is now given by the 

 formula 



the factors being adjusted to make the integral amount 

 unity. The degree of concentration varies inversely as b. 



The modified integrals with respect to m may still be 

 important when x cos ijr -f- y sin -\jr is small ; but if any pre- 

 scribed standard of smallness be imposed the range of yjr for 

 which this is satisfied becomes narrower the larger the 

 value of s/{x 2 + y 2 ). It results that the terms in question, 

 after integration with respect to yfr, become negligible at a 

 sufficiently great distance from the origin. It is not easy 

 to do more than indicate in this way the course of the 

 argument in the general case. The particular case of 

 gravity waves will be examined more in detail presently (§7). 



4. The definite integrals in (19) and (21) may now be 

 evaluated approximately by Kelvin's method |. If we 

 write, for shortness, 



k (x cos yjr + y sin ^) =/(^), . . . (24) 

 * Obtained by differentiating -with, respect to b the identity 



f Proc. Roy. Soc. vol. xlii. p. 80 (1887); Math. & Phys. Papers, 

 toI. iv. p. 303. Reference is however due to Stokes who had briefly 

 indicated the method, as an alternative to one which he actually 

 employed, in a footnote to his paper of 1850 on Airy's integral and 

 other functions. (See his collected "Papers," vol. ii. p. 341.) 



