39 Prof. T. H. Havelock. 
upper limit 0’ for 6 in order that the series for x and y may be convergent for 
negative values of yy, and there are no means of determining }’.” 
Prof. Burnside concludes that Stokes’ method cannot be used for numerical 
calculation as it is not known whether the corresponding value of 6 is less 
than the above value 0’. 
In the following notes a general method is suggested, which includes waves 
of all possible heights, ranging from the highest wave down to the simple 
infinitesimal wave. The method consists of a simple and direct extension of 
Michell’s analysis for the highest wave. The advantage is a theoretical one 
which may be expressed in this form: the parameter does not have, as in 
Stokes’ series, an undetermined upper limit, but it enters in the form 
e~**, where a may have any positive value, including zero. 
It should be stated that here, again, we have infinite processes for which no 
formal proof of convergence is given: we have to rely meantime upon a 
numerical study of the series. However, in addition, we can compare the 
method with that of Stokes for waves short of the highest; in this case 
numerical results obtained by the two methods are the same, as might be 
expected. 
Extending this comparison to the highest possible wave, we get a value for 
the quantity b’ referred to previously, that is, the value of the parameter for 
which Stokes’ series for the elevation become divergent. We obtain 0’ as 
5—)y, where 0; has the value 0:0414 approximately, or we have 0’ = 0°291..., 
the value for 0, being slightly less than the true value. 
The discussion is arranged in the following order: Michell’s form for the 
highest wave, its generalisation by means of the surface condition, method of 
approximation for the coefficients, calculation for the highest. wave, the values 
when e~** = 3, comparison with Stokes’ series, determination of b’, further 
numerical examples and remarks upon the values of the coefficients. 
2. It was shown by Stokes that the highest possible wave, under constant 
pressure at the free surface, has crests in the form of wedges of 120° It 
follows directly from his argument, as a simple extension, that the crests 
will meet at the same angle for the highest possible wave under any assigned 
surface pressure provided the pressure is stationary in value over the crests. 
Consider any assigned surface pressure of this character which is finite, 
continuous and periodic. To determine the form of the highest possible 
periodic wave, we may follow Michell’s analysis for the case of constant 
pressure up to the stage at which the coefficients are determined from the 
given surface condition. 
We might then begin with the form given in (5) below, but we may 
recall briefly Michell’s argument. Take Ox horizontal, Oy vertical and 
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