45 Prof. T. H. Havelock. 
Taking the particular case (17), we may write down one set of equa- 
tions to illustrate the type. After the first two stages, we obtain 
Iy = 0:4, kg = 7:6, bg = 4°67: using these values we find for the next stage 
the equations 
k3—0°65b3—0°432cz = 1:18 
Bi3 (O-1k3+ 2:82b3—0°543¢c3—44-25) +17:°7456—3:2168,+01 = 0 
Bi? (0:036k3—6'14b3 + 5°88c3 + 41°68) —34:688;? + 0°3328, —0:036 = 0 
B13 (0:018k3 + 0°3803—9-1¢e3— 53°77) + 2°158;? + 0:1581+ 0018 = 0 
(20) 
Eliminating /3, 63, c3, we get a cubic for 4), of which the required root is 
0:0407, the previous stages having given the values 0-0311, 0:039. Also 
from (20) we find k3 = 30, bs = 35, c3 = 40. Collecting the results to this 
stage, we find 
g = 0833, 8, = 0:0407, B2 = 00106, 83 = 0:0027. (21) 
These values are rather higher than those given by Michell (18). In 
order to determine 6; more closely, the approximation has been carried to 
the fourth stage, with the result 
yg = 0833, 6, = 00414, B2.=0°0114, 63; = 00042, 6.= 00014. (22) 
With these values, the ratio of h, the height of the wave, to L, the wave- 
length, is given by 
h/L = (velocity at trough)?/2gL 
= (1—Bi+ B2—B3+ Bs)?/24? 9m = 01418. (23) 
An interesting point about the series 1+ 8, e”*+ 6,e#*+... for the highest 
wave is the smallness of all the coefficients ®i, Bs, ..., compared with the 
first term, namely, unity; on the other hand, the numerical values obtained 
do not suggest a rapid convergence of the series after the first term. It 
appears, from the method of approximation, and from the fact that all the 
quantities Boi, Bos, ..., are positive, that successive approximations will increase 
the values of the coefficients. A test for the sum of the series, compared 
with the value of g, is obtained by considering the velocity near a crest. 
Near ¢ = 0, we have dw/dz = $¥8e-** (14+ Bi+Bot+...). 
Therefore g? = ¢7°(1+6:+42+...)? and z = 3¢7%e7/6/(14+ 81+ Bet...) ; 
and since g? = 2gy, it follows that we should have (1+ 61+ @2+...)?/g = 15. 
But with the values given in (22), this expression has the value 142, This 
is perhaps a severe test; a simpler criterion is to write down the successive 
convergents to any one coefficient; for example, those for the leading 
coefficient 6; are 0:0311, 0:0390, 0:0407, and 0:0414. 
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