Cases of Three-dimensional Fluid Motion. 359 
venient to repeat here the expressions for the velocity potential and surface 
elevation due to a cylinder and to a sphere, putting them into the same 
notation for purposes of comparison. We have, for a cylinder 
cara can, 
P Pte Bet 
+ 2¢ca? [jew du f e—*(J-) sin «(w+cw)sin(«Vu)«Vde, (21) 
0 0 
6 = 20 fla? +f?)—20 \eome du | e-Scosx(a+cw)sin(«Vu)«Vde, (22) 
0 0 
and for a sphere 
cazx carr 
= 2{ar+yP+etfype 2F+yP+e—fpp? 
d 
= ear | ee du ea I) Jo[ea/ {(a+ceuP+y?}]sin(kVu)«Vde, (23) 
0 Jo 
b= afore aye 
=a { ED, [ews [ea/ {(w+eu)?-+y?}]sin (eVu) 2Vde. (24) 
0 0 
These expressions satisfy the conditions at the free surface, namely, 
cop/ dx +96 = 0 and d6/dz = cdf/dx, when yw is made infinitesimal. Oppor- 
tunity has been taken to correct an obvious mistake in sign in the expres- 
sions for the sphere ; in the former paper, the last terms in (23) and (24) 
were given as positive instead of negative. 
Returning to the comparison with § 4, consider the expression (24) for the 
surface elevation due to a submerged sphere. The first part represents a 
disturbance symmetrical about the origin, due to a doublet at the centre of 
the sphere, together with an equal opposite doublet at a point a height f 
above the free surface. Compare now the second term in (24) with the 
surface elevation given by (6) when the pressure system is (12), so that 
f(«) = Ae-*?, The two expressions are identical, with a suitable relation 
between the constants; we must have a*= A/gp, or the corresponding 
moment of the doublet is Ac/29p. We have then two related problems. 
For the submerged sphere the pressure is constant at the free surface, and 
the surface elevation consists of the two parts in (24); the wave resistance 
depends upon the supply of energy needed to maintain the waves contained 
in the second part of (24), and this energy is supplied through the work of 
the pressure at the surface of the sphere. On the other hand, for the travel- 
ling surface pressure, 
p= gp0'T][(P?+ 07), (25) 
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