1904 - 5 .] Lord Kelvin on Deep Sea Ship- Waves. 
1075 
let A. denote the corresponding free wave-length given by the 
formula 27 rv^/g. This is the wave-length of the constituent train 
of waves corresponding to if/ = 0. For the ^-constituent, the 
component velocity perpendicular to the front is v cos i f/, and the 
wave-length is X cos 2 i f/. Looking now to fig. 26, with X cos 2 i f/ 
instead of X; and to fig. 31; and to equations (97), (98); we 
see that the portion of the depression at ( x , y) due to the 
constituent of forcive shown under the integral in (108) is 
47 r 2 bk. dif/ . 27 t{x cos if/ + y sin if/) 
X cos 2 i f/ X cos 2 i {/ 
( 110 ), 
provided x cos if/ + y sin if/ is considerably greater than JA. cos 2 t f/. 
Hence for the depression at (x, y) due to the whole travelling 
forcive, we have 
d(®, y) = 47 r 2 b 
Hf-0 
Jcdif/ 2tt(x cos if/ + y sin if/) 
X cos 2 if/ X cos 2 i fr 
(in). 
§83. The reason for choosing the limits 
^ ^ to ^ is that 
each constituent forcive gives a train of sinusoidal waves in its rear, 
and no perceptible disturbance in its front at distances from it 
exceeding half a wave-length. Look now to fig. 31, and 
consider the infinite number of medial lines of the forcives 
included in the integrals (108), (111) ; all as lines passing through 
O. Four examples, Q P, Y'Y, LK, XX' of these lines are 
shown in the diagram: corresponding respectively to i f/= ~ 0j 3 
l f / = 0,if/ = any positive acute angle, if/ = ^ . On each of the first three 
of these lines R R indicates the rear. The fourth, X X', is in the 
direction of the motion, and has neither front nor rear. The 
integral (111) must include all, and only all, the medial lines 
which have rears towards P. Hence Q P is one limit of if/ in 
(111) because it passes through P; XX' is the other limit because 
it has neither front nor rear. Thus all the lines included in the 
integral, lie in the obtuse angle POX'. Thus the integral (111) 
expresses the depression at P(&, y) due to the joint action of all 
the constituent forcives, because none except those whose medial 
lines lie in the angle P O X', contribute anything to the disturbance 
of the water at P. 
