LiiATHEM — On Two-Dimensional Fluid Motion. 2\ 



The logarithm is so defined that, as £ traverses the real axis from 

 -oo to x, the imaginary part of log (£„ - £) increases from zero to -n-. 



If Z made a complete circuit round £„ in the positive sense, log F(Z, £,) 

 would increase by 2U; hence the values of the logarithm at corresponding 



points on opposite edges of the cut differ by this amount, and the 

 integrations along the two edges combine to give 



Co 

 - R 



i j d log ff(ft 



or 2i { log £(£„) - log G{ - R) j , whose limit is 2i\ log ff(Z, ) - log K \ . 



The integral round the circumference of the cavity is of the order 

 of t' log t', and so has its limit value zero. 



On the infinitesimal semicircle round c the integral has the same limit as 



7T- 1 log F{c, l ) \ d log £ -cf lp , 

 namely 2ip log F (c, £ ). 



On the semicircle of radius 11, for H ->. oo, log /''(^', £ o* log£; and 

 if Z = Bex$i0, d£ = iZdQ. If ^(£) = K{ 1 -t /(?)}, where /(£)-»- 0. 

 then d log (o(Z) «*» i£/'(?) dd. Hence, if it be assumed that the order of 

 stnallness of /'(£), for li or | c, \ great, is such that BlogR/'(Z) ->■ *•'• 

 the integral round the great semicircle has zero limit. The assumption is 

 justified if /(£) be of the order of smallness of any negative power of »'. 

 as, for example, if //(<-,) be regular at infinity. 



The real axis contributes a line-integral in which £, being real, maj be 

 replaced by £. This integral would generally be semi convergenl as regards 

 the infinity at c, but the semi-circular detour Leads to the Cauchy principal 

 value. It will therefore be understood in what follows that den\ed integrals 

 which appear to be semi-convergent have then Cauehj principal values. 



[4*j 



