CONFOEMAL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 447 



For example consider 



( ^ s = A(w-k l )(w-k 2 ) + E(w-i)(w 3 -c^, ..... (17) 



where A > 0, B > 0, c > k t > I > k 2 > -c. 



For w real and greater than c, or for w real and less than c, ^ 8 consists of the 

 sum of two positive quantities. Hence ^ 8 has no real zeros. The angular range of 

 ^ is 2-7T, the vector angle being zero for iv : c, increasing, as w diminishes, to \v for 

 w = ki, then to ?r for w I, then to |TT for w k 2 , then to 2?r for w = c, and 

 remaining at 2x for w < c. The order at infinity is 2, so that 6 = 0. Hence 

 ^ 8 has no zeros in the relevant half-plane, and is a conformal curve-factor. 



If the angular range wei'e different from TT times the order at infinity would not 

 be zero, and the function would not be free from zeros. Tims if in ^ the order of 

 magnitude of the constants were c > I > L\ > /- 2 > c the angular range would be 

 zero and ^ would not be a curve-factor. 



It may therefore be taken as an established theorem for all curve-factors having 

 a definite order at infinity that the angular range is equal to TT times the order at 

 infinity. 



12. Curve-factors having Branch Points of Unequal Orders at the- Extremities 

 of the Linear Range. Consider the function 



ff 9 = A(u<-) + B(;-a) < '(tt--&) 1 -" ) ....... (18) 



where A > 0, B > 0, a > k > 1>, and 1 > a > 0. 



For w real and greater than a this is real ; for w real and between a and b it takes 



the form 



...... (19) 



having a vector angle which is zero for w = a and increases to x for w = b ; for 

 w real and less than b it takes the form 



-A(k-w)-~B(a-wY(b-w) l - a ....... (20) 



Thus if* ^ 9 is a curve-factor it has linear range a to b, and angular range TT. Since 

 both terms of (18) are positive, both terms of (20) negative, and the terms of (19) 

 one real and one complex, ^ 9 has not any real zeros. Since the angular range is TT 

 and the order at infinity unity, the theorem of the previous article shows that ^ fl has 

 no imaginary zeros in the relevant half-plane. Hence ^ is a conformal curve-factor. 



HYDRODYNAMICAL ILLUSTRATIONS. 



13. The utility of the conformal curve-factors so far considered may be exemplified 

 by employing them in the specification of some cases of two-dimensional liquid motion. 

 In such applications the interpretation of w will be that implied by the relation 



w = 



VOL. COXV. - A. 3 O 



