"188 T. H. Havelock 
Substituting from (36) and (43), and collecting the various terms, we 
obtain 
M/M, = 1+¢, p+. p?+c, p> +c, p*+..., (53) 
¢, = 2rysin 8, 
Cy = 3r2sin?@+ 47,(1+4sin?6)—7}, 
C, = 4rg sin? @ +797,(3 sin 0 + 2 sin? 4) 
— ir,(sin 0 — 8 sin’ @) — 2ryrj(3 sin 0 — 4sin3 0), 
C, = Srgsint 6 + 13727, sin? 6 — 2ryro(sin? 6 — 3 sin* 0) 
+4r2(1+ 14sin? 6— 12 sin*@) — 47r,(1— 8 sin*0) 
— 3r27rj(5sin? 0 — Ssin* @) —7,7{(1 + 10 sin? 6 — 20 sin* 0) 
+772(1+ 6 sin? 6— 16 sin? @) + 474(1— 4 sin? 6). 
When 0 and d are made infinite, this reduces to the expression for a semi- 
infinite stream with an upper plane boundary, namely 
1/2 1 /[2p\2 
M/M, = 1+5(=) sind +35( 2 (1+ 10sin?@) 
5 (2p\?. es 1 /2p\* ates se 
+f waa(—") (sin + 4sin 0) s5(" (1—14sin?9—40sin*@)+.... (55) 
There is also a similar reduction for a lower plane boundary. 
With a = b = id, the mid-point of the plate being midway between the 
walls, we have, from (49), 
Es 1 (2p\? ok 
ae {{4ON> : ; 
- su) (11—174sin?6—170sin*@)+.... (56) 
For the general case, we have (54) with the coefficients given by (46). 
As a numerical example, we obtain the following values: 
G = 10°; 2n/d = 0-2 
a/d 0-3 0-4 0-5 0-6 0-7 
M/M, 1-059 1-030 1-010 0-994 0-977 
FLAT PLATE BETWEEN FREE SURFACES 
9. These results may easily be modified to give approximate expressions 
when the stream is bounded by parallel free surfaces. At a free surface the 
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