The corresponding equations for the coefficients of e 2 are still not too unwieldy 

 to write down: 



A^ 2 = 0. (12) 



r]i x (x,z,t)<pi x (x,0,z,t) + rjuvu — rji<pi yy — <pz y + t]it = 0. (13) 



gr} 2 (x,z,t) + (p 2t (x,0.z,t) + m<Pu y + % (grad ^i) 2 = 0> (14) 



After eliminating ^ and rj 2 from the last two equations and performing some manipu- 

 lations one obtains a counterpart to the condition for ^ony =0: 



a 



g<P2y(x,0,z,t) + <put = (grad ^i) 2 + <p lt {<p lyy + g-tyuty). (15) 



dt 



Conditions for ^ ? , / > 2, will not be given explicitly. However, inspection of 

 the procedure shows that 



A<Pi = 0, (16) 



gvi(x,z,t) + <p it (x,0,z,t) = AilnfaO&t), ■■ ■ , <pi-i(x,0,z,t)}, (17) 



g<p iy (x,0,z,t) + <piu(x,0,z,t) = Bi{ Vl (x f 0,z,t), • • • , ^_i(x,(W)}, (18) 



where y4; and B { are certain functionals of the functions in braces. It may be noted 

 that after the ip x , . . . , ^j_ x have been found, the equations determining <p t are the same 

 mathematically as those for the linearized problem of wave motion when a prescribed 

 surface pressure p (x, z, t) — ( p 01 + e 2 P 2 + ... is given. In this case the boundary 

 condition at y — is <p lt{ + g^y + p- l p oit = 0, and 7 j 1 is given by g 7}l = 

 — fit(x, 0, z, t) — p^Poi (x, z, t) . For the case of steady motion, this problem has 

 been investigated by Hogner [1] and others. However, this observation is not of great 

 practical value since the expressions for A x and B, become rapidly more complicated 

 as i goes beyond 2. 



If this method is used to obtain the form of plane progressive periodic waves 

 with e = a/\ (= the ratio of amplitude to wave length) and with constant velocity 

 c = c + ec 1 + € 2 c 2 + . . . , one finds 



77 = A[e cos kx + y 2 Ake 2 cos 2kx +•••], k = 2x/\, (19) 



where A can be determined so that the amplitude is a. One also finds, as expected, 

 c — (gX/l-n-)^ and c x = 0. A further step in the approximation would show 

 c 2 ^0. 



II. Rectilinear Motion of a Thin Ship 



Thin Ships. Let us now suppose that the fluid contains a ship which can be 

 described in a coordinate system fixed in the ship by the equation z = ±£(x,y). We 

 shall assume that the ship is moving in the positive x-direction with velocity c(t). 

 The condition (4) of part I becomes 



£ x (x - Mt)dt,y)<p x (x,y,?(x - f*c(t)dt,y),t) + ? y <p y - <p, - cf x = 0. (20) 



We must now select a parameter e - Let 2b be the beam and / the length of the 

 ship. Take e — b/l and write £(x,y) in the form £=e£ 1 (*,y). the function £, x (x,y) 

 (with 1 for both length and beam!) is fixed and we consider the family of forms e£ x (x,y). 

 Clearly, as e^0, one approaches the problem of a flat plate of the form of the center- 

 plane section moving lengthwise through the water. This will cause no disturbance of 

 the water, so that the choice of e fulfills the requirements of part I. This choice of e 

 also defines a "thin" ship, sometimes known as a Michell ship, namely, one which can 

 be represented in the form ^ — eCi(x,y) with small values of e- 



Equation (4) may now be expanded in powers of e, just as for (1), (2), and 



111 



