444 DE. J. G. LEATHEM ON SOME APPLICATIONS OF 



If the attempt be made to construct other curve-factors of the type of a rational 

 function plus a square root, it appears that for a simple curve-factor the root sign 

 must cover only a quadratic in w, with real factors. Other factors, if imaginary, 

 would introduce branch-points ; if real they would introduce fresh corners of angle TT, 

 and so effectively yield a multiple curve-factor instead of a simple one. Thus the 

 type must be 



where f and g are rational functions. If this is to be free from zeros in virtue of 

 neither it nor its conjugate surd having zeros, it is necessary that 



{f(iv}} 2 {g (w)} 2 (iv 2 c 2 ) = const. 



If g(w] is of the first order the rationality of f(w) requires g (w) to be simply w, 

 and then the curve-factor is 



which has the angular range 2?r. This is not really a new result, for ^ = ffli- 

 Similarly, if g (w) be assumed of the second order, the rationality of f(w) restricts to 

 the form 



which is not a new result, since ^ = 



It seems therefore safe to conclude that the semi-circular curve-factor is a very 

 special type which does not admit of generalisation. 



8. The Curve-factor of Semi-elliptic Type. 'The factor which occurs in trans- 

 formation (2) above may be called the curve-factor of semi-elliptic type. It is 



^ 4 = ?rsinh + (^ 3 c 2 )'' 2 cosh a (7) 



This has, for c > w > -c, the vector angle tan' 1 [(c 2 -; 2 )' cosh a/w sinh a], and its 

 angular range is -w because the rational part of ^ 4 vanishes and changes sign at a 

 point in the linear range. 



^ 4 is free from zeros, but the reason of this is different from that which holds in 

 the case of ^,. Denoting the surd conjugate to <^ 4 by Sf t , it is seen that 



^A = c 2 cosh 2 a w 3 , (g) 



and this has two real zeros. It has to be shown that both these zeros belong to 9 te 

 so that ^ 4 has none. 



For w real ^ 4 has three different forms, namely 



(\ \ 

 >v -^ c )> w sinh a+ (w 2 c 2 ) 1 ' 2 cosh a, 



(c > w >-c), w sinh a + i (c 2 -iv 2 )^ cosh a, 

 (-c > w), w sinh a- (tv 2 -c 2 )' 1 ' cosh a. 



