Periodic Irrotational Waves of Finite Height. 48 
It is unnecessary to carry the calculation further to show the numerical 
agreement between the two methods for waves short of the highest. It 
may be noticed that in the above comparison we have gone up to the 
coefficient 83 of the present method; to obtain the agreement shown above, 
we have had to use the Stokes’ series as far as the tenth order in the 
parameter. 
7. From the comparison between (27) and (28), we see that, for waves 
lower than the highest, we are in effect dealing with a Stokes’ series whose 
parameter has the value 1e~**—8;. If we applied Stokes’ method directly 
to (27), we should obtain Ag, A3,..., in the usual way as power series in this 
parameter, and the quantity e~ 7 would be a superfluous arbitrary parameter. 
On the other hand, the present method gives a definite value of ®; for an 
assigned value of «, or theoretically gives a functional relation between 
Bi and a The method definitely connects a wave of any height with the 
highest possible wave, and any possible wave-form is given as one of a family 
whose limiting curve has crests consisting of wedges of 120°. 
Consider the expansion from the form (26) to the corresponding Stokes’ 
form (27) or (28). Assuming the convergence of the series with the 
8-coefficients, the expansion is valid over the whole range of @ for all 
positive values of «, excluding zero; it is also valid for « zero, with the 
exception of the points ¢ = az, n integral. In other words, the comparison 
confirms the view that Stokes’ series for the elevation is valid throughout, 
with the exception of the actual crests of the highest possible wave. 
We can now estimate the limiting value of the Stokes’ parameter b for 
convergence at the crests. Todo this, we compare the series (27) for the 
highest wave with a Stokes’ series, for points other than the crests. 
For the highest wave « = 0, we found 
Bi = 0:0414, B2 = 00114, B3 = 0:0042, Bs = 0:0014. 
Hence the expansion should be a Stokes’ series with the parameter 
4—(0:0414, or say 0°2919. Making the comparison between (27) and (29) 
with these values, we find 
A; = 0:2919 ; $A, = 0:0993 ; 4A3 = 0:0528 ; 
—CQ; = 0:2919; C,=0:0914; —C3 = 00429. (32) 
The agreement is sufficient to justify the comparison, when we remember 
that the @-coefficients have only been determined to the fourth stage, and 
further, that the Stokes’ series (29) for the C-coefficients presumably converge 
slowly in this extreme case. 
It should be remarked that we do not gain information from this com- 
parison about the convergence of the Stokes’ series for the separate coefficients 
142 
