Lkathkm — On TtBO' Dimensional Flniil Motion. 



25 



of £are taken the same as in articles 8 and 9, save only thai the infinity of 

 £(Q is taken at the origin and as of the type £''. The integrals 



-' log/(?, ?„) d log Gil 



and 



L-'iog^^.^rfiog^ ro 



are evaluated round their respective contours, it being known that the result 

 in each case is zero. The results are 



Tx log { G{l,W\ + if log/(0, ?„) + : log/ (5, ?„) (d log 2 + W Y ) = 0, 



(29) 



1 



IT 



if log </ (0, £„') 



log </(H,£o') (<Hog 5 + irfy) = 0. 



(30) 



In the latter formula every complex is replaced by its conjugate, and it is 

 remembered that in both formulae d% = (I for t, < 0, and d logg = for 

 £ > a. Then the elimination between the two results of the part of the 

 integral which involves d log q gives 



log{£(Q/K} = plogf(0,Z l) ) + 



which is equivalent to 



IT 



1 = 



l°g/(£. ?,), 



(31) 



{=« 



r{ r o) . jri _ "*- '^ - a >V Ex P . f^ iQ.1- ("-#-'•«•-«)»( 



a* + i (So - ay) J 7T I (a- £)* + *(£<>- a)* J 



(32) 



the integral being definite in virtue of the functional relation between \ and 

 £ which subsists at the fixed boundary. 



Formula (32) is the general formula for G{t>) which has been aimed at. 

 and its generality is now demonstrated. It is to be noticed that, for £ real 

 and a > k > 0, [f (k, ?))" has the following properties: (ii for £ > a the 

 modulus is constant, being equal to unity; (ii) for a > £ > k the vector 

 angle is constant, being mr; (iii) for £ < k the vector angle is zero. 

 Formula (32) is equivalent to formula (94) of paper D, $ 38, and represents 

 a passage from a rectilineal polygonal obstacle to a smoothly curved obstacle 

 as a limit. 



12. Tf attempt were made to approximate I" the definite integrals of 



formula (25) or (32), for an assigned form of sn tlih curved obstacle, by 



replacing the integral by a series, the resulting specification would be thai "f 



