454 DK. J. G. LEATHEM ON SOME APPLICATIONS OF 



18. If the transformation (29) be reduced to the approximate form (30) it 

 appears that 



' 



2(p + q) p + q (p + q 

 Thus the condition for the absence of the source-term is 



. . . (42) 



........ (43) 



X/ 



and the effective inertia of the ship (S being zero) is 



O T/ f 2 P<L _ o /I _ 



2 " 



The values of the parameters must of course be such that c > k > c, and it is clear 

 that there is a wide range of values of a, 73, g, and X which permit of this being true. 

 It may be well, in connexion with (44), to recall the fact that c is of the dimensions 

 of a length multiplied by a velocity, and that for a given configuration in the z plane 

 C/V is independent of V. 



Though the formula (44) does actually contain the law of the dependence of the 

 inertia of the ship (having a certain type of shape) upon the angles 2p?r and Iq-w at 

 bow and stern, it must not be assumed that the functional law corresponds to the 

 explicit appearance of p and q in the formula. The other parameters are to be 

 regarded not as data, but as functions of what ought properly to be the data, for 

 example the length and breadth or the length and area of the ship. If X and cV" 1 

 were expressed in terms of such data the formula would show explicitly the required 

 law, and would probably be a much more complicated function of p and q. 



The specification of the shape of the ship can without difficulty be reduced to 

 quadratures by putting w real on the right-hand side of formula (29), and equating 

 the real and imaginary parts respectively to dx and idy. This gives x and y as 

 integrals of certain functions of a real parameter w, but the functions are generally 

 so complicated as to make evaluation of the integrals difficult. 



In the special case of a , q = p, if dz be equivalent to da exp (ix) for w real, 

 it is seen that 



X = -pir + 2p tan' 1 {(c 2 -w 2 y''/\w}, 



ds = {(\ 2 -l)w 2 + c 2 } 

 so that, if the radius of curvature be R, 



E = _ 2* = {(^ 

 d x 



c\'ooB 1 -( x /2p) 



