RESISTANCE OF A SUBMERGED CYLINDER 421 
This result may be confirmed by using the pressure condition at the free 
surface, when the axes are moving parallel to Ox with velocity c and 
acceleration é. For these relative coordinates we have 
p CoC OME 
fh as a 1¢2_gy, 7 
Bae ae he) (7) 
_ 106 céd 
tic cramdion: (8) 
The condition that p is constant on the free surface leads to the condition 
GD. CD pa GOD @ [dw 
= 9) — - = 7]. 
Re| ap OO apa 4) dz at (Z| Co OE (9) 
It may be verified by direct substitution that (5) satisfies this condition. 
3. Suppose a circular cylinder, of radius a, centre at the origin, is 
moving horizontally with velocity c. We assume that the potential can 
be expressed as an infinite series of terms like (5) for integral values of n; 
and the quantities P, are to be determined from the boundary condition 
on the circle |z| = a. If we write 
F(k,t) = » (—i)"P,(t)"2/(n—1)}, (10) 
we have the general expression 
w= SP, (te— [ B*(1, the-He-2F die — 
0 
t (oo) 
—igt { P*(«,7) dr | Bete-tutich dic. (11) 
0 0 
We may expand the second and third terms in positive powers of z in the 
neighbourhood of the circular boundary, and we get w in the form 
We » (a Bes Qn 2”), (12) 
with 
Qn os \ P(e, eS die 
0 
(<iyrragt f ; 
a \ IT) 0 | Liet+4e-2S de, (13) 
0 0 
The boundary condition on the circle gives 
P, = ca?+a?Q¥; JE, = POG. (14) 
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