1904 - 5 .] Lord Kelvin on Deep Water Ship-Waves. 
571 
Hence finally 
J= c(j + $)ei+* | 
/ • , i \/3 i ,1 4 - 2Je cos 10 + e 
cos o + i)0 log hrr; 
1 - 2^/ecosi^ + e 
+ sin (j + i)& ta n-lVesinjg 
} . ( 86 ). 
For our present case, of 8 = J, (82) gives 
ecos# e 2 cos 20 
„+7TX + ^r 
2 J 2 J 2 
e 7 cos jO 
(87). 
With c/ and ^ thus expressed, (83) gives the solution of our 
problem. 
§ 46. In all the calculations of §§ 46-61 I have taken e= ’9, as 
suggested for hydrokinetic illustrations in Lecture X. of my 
Baltimore Lectures, pp. 113, 114, from which fig. 12, and part of 
fig. 11 above, are taken. Besults calculated from (83), (86), (87), 
are represented in figs. 13-16, all for the same forcive, (74) with 
e—‘9, and for the four different velocities of its travel, which 
correspond to the values 20, 9, 4, 0, of j. The wave-lengths 
of free waves having these velocities are [(77) above] 2a/41, 
2a/19, 2a/ 9, and 2 a. The velocities are inversely proportional 
to ^41, ^/19, J 9, J'2. Each diagram shows the forcive by one 
curve, a repetition of fig. 1 2 ; and shows by another curve the 
depression, d, of the water-surface produced by it, when travelling 
at one or other of the four speeds. 
§ 47. Taking first the last, being the highest, of those speeds, 
we see by fig. 16 that the forcive travelling at that speed produces 
maximum displacement upiuards where the downward pressure is 
greatest ; and maximum doicnward displacement where the pressure 
(everywhere downward) is least. Judging dynamically it is easy 
to see that greater and greater speeds of the forcive would still 
give displacements above the mean level where the \down ward 
pressure of the forcive is greatest, and below the mean level where 
it is least; but with diminishing magnitudes down to zero for 
infinite speed. 
And in (75) we have, for all positive values of J<1, a series 
always convergent, (though sluggishly when e=|l,) by which the 
displacement can be exactly calculated for every value of 0. 
§ 48. Take next fig. 15, for which J = 4J, and therefore, by 
