41 Prof. T. H. Havelock. 
that it is possible for some assigned stream-line below the surface, say the 
line y = a, to be a line of constant pressure. Thus we shall determine 
di, b2, ..., directly by applying the condition for constant pressure to (5) when 
y =a. The surface y =a will then be a possible free wave surface, and 
free waves will be given by assigning any value to @ in the range zero to 
infinity ; thus, by working down from the highest possible wave, we include 
in one scheme free waves of any permissible height. 
The condition that the pressure is constant for yy = « is 
g? = 2gy +constant. (6) 
It is convenient to use an equivalent form obtained by differentiating (6) 
with respect to ¢, namely 
ah = ye. (7) 
This has to be used when 
GY = {—isin(p tia) BAG" (14 Bett Bret 4...), (8) 
where Bi = bye 4, [By S WaxI-S, 0600 
Multiplying by the conjugate complex and squaring, we obtain 
qt = e 48 (sinh? « + sin? )7? (Dp + 2D1 cos 26+ 2D2c0s4h+...), (9) 
where 
Do = 14461? + (282+ B12)? + (283+ 281 B2)?+ ..., 
Dy = 281+ 28; (282+ Br") + (282+ Bi’) (283+ 28182) + ..., 
Dz = 282+ Br’ + 281 (283+ 28iB2) + (282+ Br) (284+ Bo? + 2RiB3)+..., 
Dg = 283+ 28182+ 2Bi (284+ Bo? +2BiB3)+ ..., 
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Differentiating (9) with respect to ¢, we can take out a factor 
(sinh’a+ sin)", and can collect the other terms into a sine series in 
even multiples of ¢. However, we take out also the common factor sin ¢, 
because we then have dq*/0 in a form which reduces directly to the proper 
form for the highest wave (a = 0), and, in addition, we find that the 
numerical calculations converge more rapidly. After some reduction, we 
obtain in this way 
age 4 
a $e 42/3 sin (sinh? « + sin? d)-¥3 (Ai cos f+ Azcos3p+...), (10) 
