474 DR. J. G. LEATHEM ON SOME APPLICATIONS OF 



so that, denoting this particular case of ^3., by ^ M 2p , 



11 i(i+w\ 11 ( w tt\ n it 



lt( >(w-a)' t( ~ > e-* 1 *'- ! ' J * ..... (104) 



By way of checking this result it is to be noticed that, for w real and < w < a, 



u takes the form 



a + 1D fi^iji 1 1 __ ] , . / n in \ 



(a^+w l '') aa (a-w) 4 " e- 1 '"' ~ 'e ^ la ' , 



iii which the vector angle is ill ) TT and increases from zero to XTT as w diminishes 



4 \ a I 



from a to /ero. 



For ' real and negative the form of ^ :M is 



whose modulus is ('6 iv)' !l , and vector angle x, where 



, , / w\ la i/' ''''- 

 -tan- 1 



2 \ rt / \ a / \ a/ 



it is readily seen that, as in decreases from /ero to =, x passes from \ir to the limit 

 value -OTT. 



Thus ^ is a double curve-factor of angular range ^TT equally divided between the 

 two parts of its linear range oo to 0, and to a. 



On the former part of the linear range the modulus is (a w) 1 '', so that the 

 transformation 



dz = A.(w-aY 1>f ^-M p dw ........ (105) 



gives a configuration as in fig. 14, with a free stream-line tending to parallelism with 

 the undisturbed stream. 

 In general 



flog {''.+ (w-a+ K ) J '*} F ( K ) ch = [F ( K ) log {*'''+ (,,-a + ^'^T-i f - . F ^^ x 

 - e J e ~ J e /c (2 (w a + /cj' 2 



An example, simpler than the former, is got by putting F (*) = (*/a)' /;! , so that 



/W = *M- lfc . 



This makes the integral with lower limit zero equal to 



log (a 1 /. -HP'/,) -a- 1 '' {w 1 "- (w-a) 1 ''}, 

 and gives the curve-factor 



-a- >l '{iv l "-(w-a)m ..... (106) 



with the same sort of properties as <& M . 



