1204 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



We see that the trajectories are periodic in both d and xj/ with a period 

 of 27r, and only a single region, say 



^ ^ 27r, 



^ ^ ^ 27r, 



need be considered. It is apparent that the straight lines 



(1) ^Z' = 0, e = a; (2) rP ^ 0, ^ « + tt; 



(3) \p = T, e = 2ir - a\ (4) lA = TT, e = TT - a; 



are solution trajectories which contain all values of T. Along other solu- 

 tion trajectories in this region one easily verifies that 



w 



^Dt 



XI. r vV - ^'" V '" 



3Xp / 



^^P /«„ A[cos2 xp(d) sin2 _^ gii^2 ^ (^-)p cos ^(g) gji^ q _ ^Qg ^^ 

 where \}/ = i^(0) is obtained from the solution of 



d\l/ _ [cos 6 — A(cos^ }// sin^ 6 + sin^ \l/)' cos i/' sin 0] cos 6 

 dd A(cos^ 4/ sin^ 6 + sin- i/')^ sin ^ 



From the definitions of yp and 6, it follows that the lines (3) and (4) arei! 

 physically identical with lines (1) and (2), and represent straight-line 

 la>ang and recovery respectivel3\ Likewise, the expression for T shows 

 that any non-straight line trajectory with zero bottom tension is bounded 

 by PcTVV^- Hence, as in the case of the two-dimensional model, we con- 

 clude that if the tension is somewhere greater than pcV^/w and the bot- 

 tom tension is zero, the only possible stationary configuration is the' 

 straight line lying in the plane of the resultant ship velocity and gravity 

 vectors, and making the critical angle a with the horizontal. 



F.2 Perturhaiion Solution for a Uniform Cross Current i 



At the outset we assume the tangential drag force to be zero. This 

 gives by (49) 



T = w{h + r,), (107) 



where h is the total ocean depth. Furthermore, we take pcVc to be zero.. 

 If the angle (f (Fig. 26) is small compared to unity, we assume that 6 

 and \l/ will vary only slightly from the values they would have if the i 



