with Logarithmic Singularities and Free\Boundaries. 195 



has constant modulus, (iii.) There may occur in the denomi- 

 nator of d^/dw a factor £ + 6, where b is positive. This 

 vanishes at a point in the linear range, and it is impossible 

 for a proper curve-factor to have a modulus which vanishes 

 in its linear range. Hence the only way of maintaining 

 constancy of modulus of dz/dw, so far as this element is con- 

 cerned, is to introduce into £f a similar factor f+fr. This is 

 a corner-factor, and introduces a corner of internal angle 2ir 

 into the free stream-line ; in fact it represents a cusp in the 

 free boundary, pointing inwards to the field. If b were 

 zero one branch of the cusp would belong to the fixed 

 boundary. 



Thus % is completely determined as a product of powers 

 of terms of the type of ^,3, ^ 1? and (£+6). 



If the free boundary does not extend to infinity the linear 

 range of ^must be finite, and may be taken from £= —c to 

 £=c ; and the synthesis of £f is as follows : — (i.) If a factor 

 (f— a) n , where <z' 2 >c 2 , occurs in FdQdw, put a = c cosh 7, 

 and note that 



Z? e = Z-ce-> + (f-c*) 1!2 (12) 



is a curve-factor whose modulus in its own range is 



{2c<T v (c cosh 7 -£)} 1/2 . 



Thus &^ 2n (£— a) n has constant modulus, and |f^ 271 is the 

 proper factor to be introduced into If. A particular case of 

 ^ is %? 1 = %+(£ 2 — c 2 ) 1/2 , whose modulus in its own linear 

 range is constant. The apparent possibility of introducing 

 into % any arbitrary power of ^ might seem to indicate 

 indeterminateness of <^, but the power of ^ is n °t arbitrary, 

 being determined by consideration of the angular range of 

 the (z, J) transformation, (ii.) A pair of conjugate complex 

 factors {(?-— a) 2 4 fi 2 \ occurring in d£/dw is counterbalanced 

 by introducing into If a suitable power of a factor of the 

 type 



r 7 =s(f-£) cosh 7 + (£2_c 2 ) 1/2 s inb 7 . . . (13) 

 The parameters are determined by the relations 

 cosh 7 =[{(a + c) 2 + /3 2 P /2 +{(a-c) 2 + /9 2 P /2 ]/2c, k = a/cosh 2 7, 

 and the modulus of #^ in its own linear range is 



{(f-«)3+/3 2 } 1/2 . 



(iii.) If the denominator of d^jdw contains a factor f — 6, 

 where c 2 >6 2 , the same factor (a corner factor) must be intro- 

 duced into & This indicates an inward-pointing cusp in 

 the free boundary. 



P 2 



