1073 
1904-5.] Lord Kelvin on Deep Sea Ship-Waves. 
for d(0), let us now take e = 1 . 
(-iyd(i8o°)=^ + i-J+i . 
This reduces (105) to 
, (-ly-* 1 1 (-iy 
+ 2 / — 1 2 2/+1 
(106). 
Lastly take j an infinitely great odd or even integer, and we 
find 
d(180°) = (-iy- £ (107). 
Kow fig. 26 is, as we have seen, found by superimposing on the 
motion represented by fig. 29 an infinite train of periodic waves 
represented by - 
Jh . sin 
27 TX 
and therefore h = 7r, which proves 
(97). 
§ 80. To pass now from the two-dimensional problem of canal- 
ship-waves to the three-dimensional problem of sea-ship-waves, we 
shall use a synthetic method given by Rayleigh at the end of his 
paper on “ The form of standing waves on the surface of running 
water,” communicated to the London Mathematical Society in 
December 1883.* In an infinite plane expanse of water, consider 
two or more forcives, such as that represented by (95) of § 66, with 
their horizontal medial generating lines in different directions 
through one point O, travelling with uniform velocity, v, in any 
direction. The superposition of these forcives, and of the disturb- 
ances of the water which they produce, each calculated by an 
application of (100), (101), (102), gives us the solution of a three- 
dimensional wave problem ; which becomes the ship-wave-problem 
if we make the constituents infinitely small and infinitely numerous. 
Rayleigh took each constituent forcive as confined to an infinitely 
narrow space, and combated the consequent troublesome infinity by 
introducing a resistance to be annulled in interpretation of results 
for points not infinitely near to O. I escape from the trouble in 
the two-dimensional system of waves, by taking (95) to express 
the distribution of pressure in the forcive, and making b as small 
as we please. Thus, as indicated in §§ 79, 73, 76, by taking 
b = 10^A./(10 4 .7t) we calculated a finite value for d(0). But for 
values of x, considerably greater than half a wave-length, we were 
able to simplify the calculations by taking 6=0. 
* Proc. L.M.S., 1883 : republished in Bayleigh’s Scientific Papers , vol. ii. 
art. 109. 
PROC. ROY. SOC. EDIN VOL. XXV. 
68 
