778 
1 en hd / 4 S 9 n . . . 19 
(w is the volume enclosed by S; n is a normal of unit length to 
dS). The friction has therefore no direct influence on J. 
This formula is related to that used by von Karman in the cal- 
culation of the resistance experienced by a cylinder’). 
Il. In § 3 and § 4 the moving body was replaced by a fluid 
mass with a system of forces Xj,X//. The forces Ky, are surface 
forces about which we supposed that they might be replaced by volume 
forces. This substitution will be considered more in details for the 
case of a body with a translatory motion; moreover we shall assume 
for the present that this motion is uniform, so that K;=0. 
dUn Ov, 
dn’ On 
sure p are on the contrary generally discontinuous. The normal 
component of the surface force F, is equal to the pressure p, of 
the fluid on the surface; the tangential component F, has the 
Along the surface v is continuous’), also ) and the pres- 
Ov; 5 . 
value: — u a Let us now consider two surfaces 6; and o,, the 
n 
0 
first just at the inside of the surface of the fluid that replaces the 
body, the second just outside it, so that their mutual distance 
e is small. Afterwards both surfaces must approach the surface 5, 
of the body. In this “transition layer’ we replace p and vw, by the 
continuously changing quantities p’ and »,’, so that on o; and o, 
p’, vr’ and the derivatives of v;’ up to the third order inclusive are 
equal to p, v‚, ete. (p’ and the derivatives of v;’ are zero along 6; ). 
Then the following volume forces are introduced: 
normal component: fn = an | 
n 
(20) 
tangential component : fi= —u aa 
n 
ad 
These forces are of the order «—!; integration over the depth of 
the layer gives: 
1) von KARMAN calculates the change of J from the change of the vortex 
system; by adding to this the surface integral he finds the resistance. 
2) See e.g. H. Lams, Hydrodynamics p. 572; O. Reynotps, Scientific Papers Il, 
p. 288. 
3) vi, Fi, etc. ought to be written as vectors (vi = v— n Vn); this has not been 
done here, 
