CONFORM AL TRANSFORMATION TO PROBLEMS IN HYDRODYNAMICS. 445 



The first of these is the sum of two real positive quantities, and so cannot vanish ; 

 the second is a complex whose real and imaginary parts have no common factor, and 

 so cannot vanish ; the third, because the real part has changed sign in the linear 

 range, is the sum of two real negative quantities, and so cannot vanish. Thus $" 4 

 has no real zeros, and accordingly, as no imaginary zeros are possible, has no zeros at 

 all. The change of sign of the real part of ^ 4 is an important element in the 

 argument. 



9. Modified Semi-elliptic and Semi-circular Types. Within the limits of those 

 properties by which the freedom of $" 4 from zeros is secured, namely that the product 

 of the curve-factor and its conjugate surd has only real roots and that both terms of 

 the curve-factor have the same sign for real values of w outside the linear range, 

 there is room for modification of the type ^ 4 by the introduction of an additional 

 parameter. Considering 



, ....... (9) 



it is seen that, provided c > k > c, both terms of ^ 5 are positive for real values of iv 

 greater than c, and both negative for real values of w less than c ; thus ^" 5 cannot 

 vanish for any real value of w. If the conjugate surd be ^ 6 , 



# B S = (c 2 + /; 2 shih 2 a ) cosh 2 a- (ic + k smh" a) 3 , ..... (10) 



which has only real zeros. Hence ^ 5 has no zeros, and is a curve-factor. Its angular 

 range is TT. On the other hand, if k > c, ^ is positive for w = + <x> and negative for 

 iv = c, so it must have a zero and is not a curve-factor. The same applies when 

 k < c, since then ^ 5 is negative for w = co and positive for w = c. 

 Similarly ^ may be modified to the form 



V t = w-k+(w'-G') 11 ' ......... (11) 



with the restriction c > k > c. The only zeros that might be possible are those of 



......... (12) 



and this has only one zero, which is real. Thus ^ has no zeros, and is a curve-factor 

 of angular range TT. 



It may be noted that 



- B ........... (13) 



The utility of the adjustable parameter k will be seen in the working of particular 

 examples, 



