49 Prof. T. H. Havelock. 
for higher values of the parameter; the result concerns the series for the 
elevation. We find that Stokes’ series for the elevation becomes divergent 
at the crests when the parameter has the value 0:291..., so far as the 
numerical calculation has been carried. 
In this connection reference may be made to Wilton,* who concluded that 
the Stokes’ series certainly diverge for a parameter greater than 1/e, and 
who estimated the limiting value to be in the neighbourhood of 4. 
Wilton works out in detail a numerical example with the parameter 
b = 0316, for comparison with the highest wave. According to the present 
analysis, this is beyond the limiting value for 6; the series should be 
divergent at the crest. This may well be the case, notwithstanding that the 
coefficients C, as calculated by Wilton, diminish steadily as far as the order 
shown ; since the series is supposed to be divergent only at the crests, one 
might expect the divergence to become evident numerically only after 
calculating a large number of terms. The example may serve as an illustra- 
tion of Prof. Burnside’s criticism, that it is necessary to know the limiting 
value of 6 before Stokes’ series can be used with confidence for numerical 
calculation. 
8. We may examine briefly the present method for waves of small height. 
It is of interest first to consider the exact expression 
ae = 2-8 (33) 
We can integrate dz/dw and so obtain the equation of the stream-lines 
in finite form, and also exact expressions for the variations of pressure along 
any stream-line. To find how far (33) satisfies the condition for a free wave 
under constant pressure at a stream line wy = a, it is simpler to expand first 
before integrating ; we can then express g? and y as cosine series. In this 
way we find at the wave surface y = «, writing down the variable part only, 
Const. x (q?—2gy) = {4ge**—(h ee *— gto — 185 e ®_...)} cos 2h 
+{}ge*—($ e *#—55, ce *_...)}cos 4g 
+ {ald ge 8*— (fre — phy oe—...)}008 6 
(34) 
Hence, if we take g-1=1+ 4c 4, the pressure is constant up to, 
and including, terms in e~*; and the next term is the small quantity 
—ats & cos 6d. This value for 1/g is Stokes’ expression 1+0?, the 
* J. R. Wilton, loc cit. ante. 
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