where z = x + iy and refers to a fixed point (ar,y). Thus 



c' I' 



w = - <$\ \n{z-x') dx' =^[{z-x') In {2-x') + x'] 



a a 



at 



w = <t> +iil/ =*[('2-6) In (z-h) - (z-a) In (z -a)] 

 after dropping a constant terra. Hence 



<f>=^[(x-b)lnri,-(x-a)lnT^-y(d^-dj], xji = -^U^x-'^^^a-(^-^)Qb + y^'^ -^A 

 where 



[57a,b] 



r„= [(x-a)^^y^f, r, ~[(x-h)^ ^ y^]^ , 0^ = tan"! -^ ^^ = tan"! -i^ 



X — X —a 



and 6^ and 0^ may be allowed to vary continuously without restriction. Also 



w=_^=.aln— ^ = _-r = oCg ^^j [57c,d] 



dx Tf^ dy 



The conjugate flow is that due to a uniform sheet of line vortices; the vortex strength 

 or circulation per unit of width of the sheet is 2(ra, and the circulation around the entire sheet 

 is 27rac; see Section 40. 



Wtt is negative, the sources become sinks, or the direction of the circulation around 

 the vortex sheet is reversed. 



(For notation and method; see Section 34; Reference 3, p. 81.) 



58. SOURCE SHEET IN A UNIFORM STREAM 



Let the sheet of sources described in the preceding section be immersed in a uniform 

 stream flowing at velocity V toward negative x. For simplicity let a = - c, h = 0, so that the 

 sheet, of width c, extends from x = -c to x = 0. Adding, from Section 35, a term Vz for the 

 stream, and replacing a hy gU, where g will be assumed to be positive: 



w = U \z + g [zlnz- (z + c) ln(z + c)]\ [58a] 



cf} = U\x + g[x\nr- (x + c) Infj -y(d- di)]\ [58b] 



if; = U4y + g\ xe -(x + c)dy-yln— \\ [58c] 



131 



