Periodic Irrotational Waves of Finite Height. 44 
when « = 0. The hypergeometric series in (14) can be summed in this case 
and we find 
18,/3 6r+1 
Bis gal eke ele ee 16 
apis mn 9(2n+1P—(6r+1)? eo) 
The equations (15) reduce to 
1—Bi + 4By?—28182—Bie = k (A—7 Bi— 3s B23} Bs), 
5B1—482— 281? + 108182 + 5Bi? = k (ty +35 Bi—4382— 3185 Bs), 
882+481?—7BiB2—7B3 = k(shg+ yi hit 38 B2—i% Bs), 
116182+1183 = k (stg t+a5ehit ahs Bet 4883), (17) 
with k = 189,/3/7. 
These are Michell’s equations for the highest wave. Without specifying 
any definite method of approximation, Michell states that sufficiently close 
values are given by 
g = 0832, 6, =00397, f2=00094, @3;=0:002. (18) 
In order to compare results for different values of «, it is desirable to adopt 
some consistent scheme of approximation. 
In general, in the equations (15), we substitute 
gBo = 14+ 6i+heB?+hsBi3+..., 
Bo = 281? + 03813 + ..., 
B3 = 6383+ cs8it + ..., 
sis gre Decca ce dh fe (19) 
For a first approximation, write down the first two equations up to terms 
in #1, and we get two equations from which to determine i; and f;. The 
first of these equations is, in fact, independent of #; on account of the form 
of (19). 
For a second approximation, retain the value of /, so determined, and 
write down the first three equations of (15) up to the terms in ,’, the first 
of the three being again independent of 61; from these, we determine 
ka, B2, and a second approximation to ;. For the third stage, using the 
values of fy, ke, and 02 already found, and writing down the first four equa- 
tions of (15) up to the terms in (,°, we determine ks, 63, ¢3, and a third 
approximation to §;. Using (19) we obtain the corresponding values of 
9, B2, B3,.... ab any stage. The nth approximation to , is given by an 
equation of the nth degree in 8; but there is no difficulty in practice as to 
the particular root since we follow it through from the first approximation. 
The method is simple in plan, if somewhat tedious in practice; so it is 
not necessary to give the details of the following calculations. 
