Periodic Irrotational Waves of Finite Height. 50 
parameter } having the value 4e~ to this order. It is, of course, impossible 
to make the right-hand side of (34) zero for all values of ¢ merely by 
choosing g a suitable function of « However, the fact that (33) satisfies 
approximately, to the order shown above, the conditions for a free wave of 
any height, explains the smallness of the coefficients 61, B2,... even for the 
highest wave when « is zero. 
Returning to the general equations (15), for a numerical example of a 
wave of moderate height we take e~?* = 3/10. In this case we shall only 
carry the approximation to the second stage, to illustrate the character of 
the coefficients. We have the following numerical values : 
Bo = 1:008 ; By = —8:24; Ba = —7°336; 
Bos = —0:004 ; Bis — 0:082 ; Bas = 121% 
From these we obtain 8; =—0-:0018; Bs =—0:00066. Making the com- 
parison with a Stokes’ series, as in the previous sections, we find 
A, = —C, = 01018, 4 Ay = Cy = 0:0104, 1A; = —C; = 0:0016. 
The numerical values confirm the impression that while the 8-coefficients 
diminish indefinitely as the height of the wave hecomes smaller, it is more 
difficult to obtain their values by the method of successive approximation 
used in dealing with the infinite set of equations for them. 
The behaviour of the 6-coefficients is made clearer by studying the leading 
terms Boi, Bos, ..., on the right-hand side of equations (15). Over the whole 
range for a, from zero to infinity, By, only varies from about 1:24 to 1; 
consequently, from the first equation, g is never much different from unity. 
From the remaining equations, we see that (4, G2, 83, .-., form a parallel 
series to Bos, Bos, Boz, ---, taken in order. 
It is only for the highest wave («=0) that all the terms of the latter series 
are positive and decrease steadily to zero from the first term; for other 
values, the series is not quite so simple in form, although in all cases the 
terms converge ultimately to zero. The character of these terms is best 
illustrated by numerical examples, such as are given in the following Table: 
144 
