CONFORMAL TRANSFORMATION TO PEOBLEMS IN HYDRODYNAMICS. 463 



parts of one line, and so lead to a specification of liquid flow past two objects of 

 oval form. 



Fig. 9. 



Another double curve-factor, corresponding to two curves separated by a straight 

 line, is 



where X > 0, and a>b>c>d. It lias a total angular range of TT, made up of fr 

 from each of the two curved portions. 



Fig. 10. 



It would be interesting to know whether a double curve-factor is necessarily the 

 product or 'other simple function of simple curve-factors. 



SYNTHESIS OF CURVE-FACTORS FROM ELEMENTARY TYPES. 

 30. A simple function which has not yet been considered as a possible curve-factor is 



F (>) = -.'+ (' 3 -c 2 )' /J . 



The product of this and its conjugate surd has only one zero, a real one, so F (?r) 

 need only be tested for real zeros. 

 It is to be noticed that 



jL; w _(V-c 2 VH - i w 



dw i 



which is negative both for w real, positive and greater than c, and for w real, 

 negative and less than c, due regard being paid to the significance of the square 

 root. Thus {w (w 2 c 2 ) 1 ' 2 } is positive for w real and greater than c, and continually 

 decreases as w increases, having the limit zero for >-> + > ; for w real and less 

 than c, {w (w 3 c 2 ) 1 ' 2 } is negative and continually increases to the limit value zero 

 as w-> oo. It follows that F(w) has no stationary values for w real and w 2 > c 2 ; 

 the values of F(w) for w = , w = c, w = c, and w = + oo } are k, k + c, kc, and 



VOL. CCXV. A. 3 Q 



