Deep Wider Ship- Waves. 735 



joining them experiences chances of direction which are 

 infinitely small in comparison with the radian. For liberal 

 interpretation of this condition see § 61 below. Water being 

 assumed frictionless, its motion, started primarily from rest 

 by pressure applied to the free surface, is essentially irrota^ 

 tional. But we need not assume this at present : we see 

 immediately that it is proved by our equations of motion, 

 when in them we suppose the motion to be infinitesimal. 

 The equations of motion, when the density of the liquid is 

 taken as unity, are : — 



W,0,ydj_ dp. 



■ (59), 



where g denotes the force of gravity and p the pressure at 

 (>. c, t). Assuming now the liquid to be incompressible, we 

 have 



g + §=0. ...... (CO). 



§ 37. The motion being assumed to be infinitesimal, the 

 second and third terms of the first members of (59) are 

 negligible, and the equations of motion become : — 



dt 1 dx\ ,,,. 



« + ( } ' 



This, by taking the difference of two differentiations, gives: — 



which shows that if at any time the motion is zero or irrota- 

 tional, it remains irrotational for ever. 



§ 38. If at any time there is rotational motion in any part 

 of the liquid, it is interesting to know what becomes of it. 

 Leaving for a moment our present restriction lo canal waves, 

 imagine ourselves on a very smooth sea in a ship, kept moving 

 uniformly at a good speed by a tow-rope above the water. 

 Looking' over the ship's side' we see a layer of disturbed 

 motion, showing by dimples in the surface innumerable little 

 whirlpools. The thickness of this layer increases from nothing 

 perceptible near the bow to perhaps 10 or 20 cms. near the 



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