1901 - 5 .] Lord Kelvin on Deep Water Ship- Waves. 
569 
many as the scale of the diagram allows to he seen on it. There 
is a somewhat close agreement between each of the light curves, 
and the part of the heavy curve between a maximum and the 
minimum on each side of it. Thus we see that even with e so 
small as ’5, we have a not very rough approximation to equality 
between successive half periods of the first member of (78) and a 
single term of its second member. If e is < 1 by an infinitely 
small difference this approximation is infinitely nearly perfect. 
It is so nearly perfect for e=*9 that fig. 12 cannot show any 
deviation from it, on a scale of ordinates 1/10 of that of fig. 11. 
The tendency to agreement between the first member of (78) and 
a single term of its second member with values of e approaching to 
1, is well shown by the following modification of the last member 
of (74) 
n = £c- 
« i-f) 
1 - 2e cos 0 + e 2 (1 — e) 2 + 4e sin 2 \d 
i(l-e 2 ) 
(80). 
Thus we see that if e = 1 , II is very great when 0 is very small ; 
and II is very small unless 0 is very small (or very nearly = 
Thus when e = 1, we have 
gc ' (1 -e) 2 + e6 2 
( 81 ); 
which means expressing II approximately by a single term of the 
second member of (78). 
