If one lets p = v\, one gets 



1 / M 



K{vx,vy) =— / du — e^ /y2 cos nx> (38) 



^ 2 A Vm 2 -^ 2 



Both forms occur frequently. Birkhoff and Kotik [1, 2] introduce new variables: 

 u — x—£, v = y + rj, resulting in 



4g 2 P f L f° 



R = / du dvM{u,v)K{vu,w), (39) 



7TC 2 J_ L 7_ 2 ff 



where 



min 04L,}4L—u) min (H+v,0) 



M{u,v) = J da; J dy f »(m + z,v - y)£ x (x,y); (40) 



max (— %L,— Y^li— u) max (—H,v) 



here the hull extends from — ViL to V6L and the draft is ff. The obvious advantage 

 of this form is that the data concerning the hull are concentrated in the one function 

 M which Birkhoff and Kotik call the "hull function". Finally, by changing the order 

 of integration and using simple properties of the exponential and cosine functions, one 

 may obtain 



R = / d\ [P 2 + Q% (41) 



7TC 2 Jx VX 2 - 1 



where 



P = jj dxdy $ x (x,y)e vX * y cos vXx, Q = jj dxdy £ x (x,y)e vX2y sin v\x. (42) 



s s 



If jdx\£ x (x,y)\^.A, a constant, it is easy to show that the integral for R is absolutely 

 convergent (one gets R^A 7 r-' L pL 2 A 2 c 2 ), so that the change of order of integration is 

 allowable [cf. Birkhoff and Kotik, 2, pp. 6-7, 19-21]. 



If the function £,{x,y) is of the simple form £{x,y) — X(x)Y(y), where, say, 

 Y(0) — 1 and thus X(0) = b — half the beam, the last expression for R takes on a 

 somewhat simpler expression insofar as one has to deal only with single integrals in 

 the expressions for P and Q : 



P = f dx X'(x) cos v\x • / dy Y(y)e'*\ (43) 



-}iL -H 



and similarly for Q with sine replacing cosine. Ships which may be so represented are 

 called "elementary ships" by Weinblum [10, p. 83] and have been treated extensively 

 by him, Havelock, Wigley and others. In the special case when the ship is wall-sided, 

 Y(y) zz 1, — H^y^O, and the second integral in P and Q becomes 



(l-exp-v\*H)/v\*. 



If H is allowed to become infinite, one obtains the wave resistance for an infinitely 

 long vertical strut: 



L/2 



4pc 2 r r 



R = / / dxdZX'(x)X'($)K(v(x - £)), (44) 



-Ll-i 



where 



f°° cos \x 



K(x) = / d\ === (45) 



J x x vx 2 - i 



115 



