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



The transformation must have angular range zero, and if b > c a suitable type is 



* dw, .. ...... (48) 



where ^ is a curve-factor of linear range cto-c, and angular range TT. For example 

 if the factor ^ 10 be adopted the transformation is 



- + -cw (49) 



v(\+iy+ l '*(w->' 



When this is expanded, for w great, in the form 



dz = V- 1 ( 



it is found that 



S=p C -ib 



and 



As S must vanish, the value of /,; must be 



to, ........ (50) 



and the other constants must be such that this / is between c and c. 



The inertia-coefficient, for longitudinal motion, of the system consisting of the ship 

 in liquid at rest at infinity, is 



INFLEXIONAL CURVE-FACTORS. 



21. In seeking for curve-factors whose angular range shall be zero it will be 

 observed that as w decreases through real values from +c to c the vector angle 

 of the complex 



(w) = \(w-k) + (w t -<?)\ ........ (52) 



wherein X > 0, will increase from zero through I-TT to TT, or will change from zero 

 to tan" 1 {cX -1 (k 2 c 2 )" 1 ' 2 } as a maximum or minimum and then relapse to the value 

 zero, according as X (wk) does or does not change sign for a value of w in the linear 

 range. Hitherto the angular range -K has been secured by postulating c 2 > k 2 ; it may 

 now be enquired whether, when c 2 < k\ the function F (iv) can constitute or help to 

 constitute a conformal curve-factor. 



