Deep Water Ship-Waves. 753 



19, 20 ; we see obviously that it* any two equal and similar 

 torches are applied, with a distance l\ between corresponding 

 points, and if the forcive thus constituted is caused to travel 

 at speed equal to \ ! gk^lir, being, according to (77") above, 

 the velocity of free waves of length \, the water will be left 

 waveless (at rest) behind the travelling forcive. 



§ 58. Taking for example the forcives and speeds of figs. 18, 

 19, 20, and duplicating each forcive in the manner defined in 

 § 57, we find (by proper additions of two numbers, taken from 

 our tables of numbers calculated for figs. 1#, 19, 20), the 

 numbers which give the depressions of the w r ater in the three 

 corresponding waveless motions. These results are shown 

 graphically in fig. 21, on scales arranged for a common 

 velocity. The free wave-length for this velocity is shown as 

 4 7 in the diagram. 



§ 59. The three forcives, and the three waveless water- 

 shapes produced by them, are shown in figs. 22, 23, 24 on 

 different scales, of wave-length, and pressure, chosen for the 

 convenience of each case. 



§ 60. As most interesting of the three cases take that de- 

 rived from j = 9 of our original investigation. By looking 

 at fig. 23 we see that a pontoon having its bottom shaped 

 according to the D curve from — 3</ to + 3<7, 1J free wave- 

 lengths, will leave the water sensibly fiat and at rest if it 

 moves along the canal at the velocity for which the free- 

 wave-length is ±q. And the pressure of the water on the 

 bottom of the pontoon is that represented hydrostatically by 

 the F curve. 



§ 61. Imagine the scale of abscissas in each of the four 

 diagrams, figs. 21-24, to be enlarged tenfold. The greatest 

 steepnesses of the D curve in each case are rendered sufficiently 

 moderate to allow it to fairly represent a real water-surface 

 under the given forcive. The same may be said of figs. 15, 

 16, 18, 19., 20 ; and of figs. 13, 14 with abscissas enlarged 

 twentyfold. In respect to mathematical hydrokinetics gene- 

 rally; it is interesting to remark that a very liberal inter- 

 pretation of the condition of infinitesimality (§ 36 above) is 

 practically allowable. Inclinations to the horizon of as much 

 as 1/10 of a radian (5°'7 j or, say, 6°), in any real case of 

 water-waves or disturbances, will not seriously vitiate the 

 mathematical result. 



§ &2. Fig. 17 represents the calculations of d(0°) and 

 ( — l)jd(180°) for twenty-nine integral values of ; ; 0, 1, 2, 

 3, ..*.'. 19, 20, 30,40, .... 90, 100; from the following 

 formulas, found by putting = 0° and 0=180°; and with 

 e = '0 in each case, and c = l 



