1064 Proceedings of Royal Society of Edinburgh. [sess. 
where h denotes the height of crests above mean water-level in the 
train of sinusoidal free waves left in the rear of the travelling 
forcive ; A denotes the area of the forcive curve (fig. 25) ; being 
given in § 66 by the equation 
A = irbk (98): 
and A, given [§ 39, (71)] by 
A = 27 TvPJg (99), 
denotes the wave-length of free waves travelling with velocity v. 
§ 71. A very important theorem in respect to ship-waves is 
expressed by (97). Without calculation we see that, if A is very 
great in comparison with 7 rb, (the “ mean breadth ” of the forcive 
curve according to § 66), h must be simply proportional to A, for 
different forcives travelling at the same speed. This we see 
because, for the same value of b, h/k is the same, and because 
superposition of different forcives within any breadth small in com- 
parison with A, gives for h the sum of the values which they 
would give separately. Farther without calculation, we can see, 
by imagining altered the scale of our diagrams, that hX/A must 
be constant. But without calculation I do not see how we could 
find the factor 47 t of (97), as in § 79 below. 
§ 72. The effect of the condition prescribed in § 71 is illustrated 
and explained by considering cases in which it is not fulfilled. 
For example, let two forcives be superposed with their middles at 
distance JA ; they will give h = 0, that is to say no train of waves. 
The displaced water surface for this case is represented in fig. 27. 
Or let their distance be JA or §A ; the two will give the same value 
of h as that given by one only. Or let the two be at distance A ; 
they will make h twice as great as one forcive makes it. 
§ 73. In figs. 26, 27, 29, 30, representing results of the calcula- 
tions of §§ 78, 79 below, the abscissas are all marked according 
to wave-length. The scale of ordinates corresponds, in each of 
figs. 26, 27, 29, to k = 243’89, and irb = 1*0251. 10 -3 . A. This 
makes by (98) and (97) A = JA, and Ii — tt. Fig. 30 represents the 
curve of fig. 29 at the maximum, in the neighbourhood, of O, on a 
greatly magnified scale: about 1720 times for the abscissas, and 
39 times for the ordinates. 
§ 74. Fig. 26 shows, on the right-hand side, the water slightly 
