due to a I 1 ravelling .Disturbance. 547 



nsion in ( 



n(-§) 



amended by continuing the expansion in (26) a step further, 

 and using the formula * 



i: 



•» ■ • • <"» 



The infinity which occurs when Q— ±2 7r * s due ^° * ne 

 concentration o£ the source. If we introduce a diffusion- 

 factor e~ kb in our formulae, as already explained, the second 

 factor in (43) is replaced by 



A/li 



tan 2 6 



(45) 



which vanishes at the places in question. 



7. It is possible in a definite problem such as the present 

 to estimate the importance of the terms which were omitted 

 in' the integration with respect to k in § 3. 



Putting 



cf>(k) = ike- kh /pa, <r 2 =gk, . . . (46) 



in (16), we have to consider the integral 



I ■= rr-r r, .... 47) 



Jo 1 ~9 2 ^ 2 cos^ 



taken along the imaginary axis, p being written in place of 

 jc cos ^r-\-y sin yjr, and yu, put = 0. This is equivalent to 



e-te+^mdm /AQ . 



. . . (4fc) 



-j; 



,„ 1— g~zc{im)z cosyjr' 



The important part of this integral is due to comparatively 

 small values of m, so that the second term in the denominator 

 .may be omitted. We thus obtain 



or - ,.„„..., .. 1 „i,.,...-A,. . • ( 49 > 



(p+ib)' 1 ' (d'cos ty + y sin ty + ib) 2 ' 



this being, in fact, the first term in the asymptotic expansion 

 of (48) by the usual method. The result has to be integrated 

 with respect to i/r. Since 



f ' * t = f* ^ = _ _ J«_. _ (50 ) t 



J.^tf cos^r+#sin^r + i& J_ ff rcos^ + z6 //(r 2 + 6 2 )' v y 



* Haveloek, Proc. Roy. Soc. A. vol. lxxxi. p. 422 (1908). 



t The equality of the second and third members is established by 



putting t= tan -ij, and integrating over the contour of the upper half 



of the ^-plane, having regard to the singularity at t=e i «- j where 

 <a= tan -1 (£>/>•). 



202 



