Deep Water Ship- Waves. 751 



equal positive and negative values of 6 from 0° to 40° or 50° 

 by {(83), (81)), (87)}; and for all larger values of 6 take 



d(0)=(-l)/d(18O o )sinQ-+i)0 . . (91), 



where d(180°) is calculated by {(83), (86), 87)}. This used 

 in (8$), makes D(0)=O for all positive values of 6 greater 

 than 40° or 50° ; and makes it the double of (91) for all 

 negative values of beyond — 40 c or —50°. 



§§ 55, 56. Rigid Covers or Pontoons, introduced to apply the 

 piven forcive (pressure on the water-surface). 



§ 55. In any one of our diagrams showing a water-surface 

 imagine a rigid cover to be fixed, fitting close go the whole 

 water-surface. Now look at the forcive curve, F, on the 

 same diagram, and wherever it shows no sensible pressure 

 remove the cover. The motion (non-motion in some parts) 

 of the whole water remains unchanged. Thus, for example, 

 in figs. 13, 14, 15, 16, let the water be covered by stiff covers 

 fitting it to 60° on each side of each C ; and let the surface 

 be free from 60° to 300° in each of the spaces between these 

 covers. The motion remains unchanged under the covers, 

 and under the free portions of the surface. The pressure II 

 constituting the given forcive, and represented by the F curve 

 in each cass, is now automatically applied by the covers. 



§ Di). Do the same in figs. 18, 19, 20 with reference to the 

 isolated forcives which they show. Thus we have three 

 different cases in which a single rigid cover, which we may 

 construct as the bottom of a floating pontoon, kept moving 

 at a stated velocity relatively to the still water before it, 

 leaves a train of sinusoidal waves in its rear. The D curve 

 represents the bottom of the pontoon in each case. The 

 arrow shows the direction of the motion of the pontoon. 

 The F curve shows the pressure on the bottom of the pontoon. 

 In fig. 20 this pressure is so small at —2g that the pontoon 

 may be supposed to end there; and it will leave the water 

 with free surface almost exactly sinusoidal to an indefinite 

 distance behind it (infinite distance if the motion has been 

 uniform for an infinite time). The F curve shows that in 

 fig. 19 the water wants guidance as far back as — 3^/, and in 

 fig. 18 as far back as — 87 to keep it sinusoidal when left 

 free : q being in each case the quarter wave-length. 



§§ 57-60. Shapes for Wavelets Pontoons, and their Forcires. 



§ 57. Taking anv case such as those represented in figs. 18, 



3 D 'i 



