Lkathkm — On 'Vwo- Dimensional Fluid Motion. 



23 



10. The equality of formula (20) must remain true if every term in it is 

 replaced by its conjugate complex. The conjugate complex of the G function 



has already been specified; that of log I' (£) is logT - iS. Thus the new 



formula is 



j = « 



- 2ip log F(c, Z„) + - log F(£, Z„) (d log r - w&) 



I 7T 



£ = -« { = » 



1 



log^^^)(rflogr-^) = 0. 



(21) 



Since, by hypothesis, r = A' for £'-' > «'*, rf log r = over these ranges. 

 Hence formulae (16) and (21) can be simplified, and are respectively equivalent 

 to 



?=« 



? = « 



log F(£, £,) d log r - g- j log J(5, &,) dS, (22) 





J = -co 



- - p log A(c, £,) - - log F% Z a ) d log r 



{ = - « 



i i 



2n ( J J 



{ = — CO £ = *< I — — fl 



Addition of corresponding sides of these equations gives 



logJf(f5-)^. (23) 



log 



nri 



l>logA( C) £ ) 



1 



so that 



j log F (£,?.) A 



"{ = - 8 



tftf.) - A' (JPfc «f * Exp ' - I log F(l &)< 



U4 



(25) 



It being remembered that there is necessarily a functional relation 

 between S and £, this formula is recognized as an expression of t Zo) '" the 

 form of the limit of a product of the character discussed in article 7. above 



Thus the generality of the synthesis of £(Z ) by factors of the type 

 of F(Z,Z ) is demonstrated. The form (25) is preferable to the form LO 

 since K is always real while A" is complex. In the hydrodynamical 

 application it may be convenient to take A' to be unity. 



