BY T. H. Havelock. 
Further, we have 
m2 
ly = | cos (k tan 6)dé = 4ne* 
0 
M) = — ie sin (k tan $)df = — $ {e*h (e*) — ela (e~*)}. (28) 
0 
We shall find it necessary to go as far as the sixth term in numerical calcula~ 
tion of A and B ; we therefore record to this order explicit expressions for L and 
M obtained from (27) and (28). 
Ib, == elo, 
L, = — rk (1—hk) e*, 
L, = 4rk (3 — 6 + 22) e*, 
Ly = — 4nk (3 — 9k + 642 — kh) e—*, 
L; = 75nk (15 — 60k + 60k? — 2043 +- 2h*) e-*, 
Lg = — pth (45 — 225k + 300k? — 1503 + 30k* — 2k°) e—*, 
M, = — kei (e*) +1, 
M, = k(1—h)ehi (e*) +-&, 
M,; = — 3h (3 — 6h + 2k?) e*hi (e*) + 4 (1 — 4k + 29%), 
Mz = 3h (3 — 9k + 6h? — h°) e~*hi (e*) 4+ 4k (5 — 5k + F), 
M; = — 75k (15 — 60k + 60k? — 20K? 4 2k*) e*hi (e*) 
+ af; (3 — 28k 4 442? — 183 + 284), 
Mg = sk (45 — 225k + 300K? — 150K? + 30k* — 2h>) e*h (e*) 
+ Jk (93 — 198k + 124k? — 28% + 2%). 
6. The first case we shall examine is that already discussed in § 3, a cylinder 
whose centre is at a depth of twice the radius. It has been remarked that this 
is an extreme case, but it has the advantage, as far as the calculations are con- 
cerned, of magnifying the difference between the first and second approxima- 
tions and so of lightening the numerical work involved. In the notation of the 
previous sections, we have 
f=2a; k= 2of=4rfl/ro; bh = xoar/4f = k/32. (29) 
Collecting the terms in (18), (19) and (23), the regular waves established to 
the rear of the cylinder are given by 
nla = rke~™ sin kyo — pgrke”** sin xox 
+ shak?e—** (A cos Kot — B sin kz). (30) 
The first term is the first approximation, and the amplitude in this case has 
a maximum at * = 2, or when the velocity is such that the wave-length is 2zf. 
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