194 Dr. Leathern on Two- Dimensional Fields of Flow ', 



Any distribution of electrodes, symmetrical about the 

 central line of the strip, can be dealt with by introducing 

 into w terms of the type of formula (1). 



8. Liquid flow with free stream-lines, with or without sources 

 and vortices. — When the fixed part of the boundary is a 

 rectilineal open polygon the (z, J) relation is 



dz = £f(r)F(?K, 



where F(£) is a product of Schwarzian factors determined 

 in the usual manner, and ^(£) is an as yet unknown curve- 

 factor representing the free boundary. The problem is 

 reduced to quadrature when the form of % has been ascer- 

 tained, and it will now be shown how ^is determined from 

 the condition that | dzjdw | is constant in the free boundary. 

 The field relation is of the type 



«=A*-s£iori<r— yf«+sg log {§=*=§}, (io) 



so that dwjd£ is an algebraic fraction in £ of known form ; 

 and dz/dw— &Fd%/dw. The modulus of this can be made 

 constant in the linear range of % by building up % by factors, 

 each factor being itself a curve-factor of such a character 

 that, when it is associated with a particular factor in nume- 

 rator or denominator of FdQdw, the combination has (for 

 values of f corresponding to the free boundary) a modulus 

 independent of f. Every factor of FdQdw bein^ dealt with 

 in this way, the complete product of the curve-factors is £f . 



Tf the algebraic sum of the outputs of the given sources, 

 including a possible source at infinity of output 7rA, is not 

 zero, the free stream-line will extend to infinity. In this 

 case the linear range may be taken from f=— so to f=0, 

 and the synthesis of ^is as follows : — (i.) A factor (f— a) n , 

 where a is real and positive, occurring in Fd£/dw, is to be 

 associated with &w*, where ^ 13 = a l/2 + £ 1/2 , which is known 

 to be a curve-factor of the assigned linear range with modulus 

 (a — £) 1/2 in that range, (ii.) A product of conjugate com- 

 plex factors {(?-— «) 2 + /3 2 } may occur in the denominator of 

 d£/dw, and will certainly occur in the numerator if there are 

 sources or vortices clear of the boundary. Now 



r 61 =r+(« 2 +/s 2 ) ,/2 + {2(a 2 +^) ifl +2«p /2 r 1 ' 2 (ii) 



is a curve-factor having the assigned linear range, with 

 modulus {(f-a) 2 + £ 2 } 1/2 in that range. Hence 



^i 2 {(r-«) 2 + £ 2 } 



