775 
§ 2. Impulse of a vortex system. 
The impulse of a vortex system is defined as the impulse of a 
system of forces that instantaneously can generate the given vortex 
motion in the fluid from rest '). When the fluid is unlimited and when 
it does not contain any body, this impulse is given by the formula: 
B sl {fae dy der w ee EN 
ela SO hod Caan ee cn a, a ME 
(9 = density of the fluid; r is the radius vector of a point 2, y, z; 
w is the vorticity, defined by w=rotv; C; is the circulation 
round a vortex line; A; the surface enclosed by the line, regarded 
as a vector). *) 
§ 8. Elementary derivation of the formula for the resistance. 
Let us consider a body in an unlimited fluid; originally all be 
at rest. By forces acting on the body it is j set into motion; let us 
have for the moment é: 
f = resultant of the forces acting on the body; 
B= the eee or momentum of the body =o’ 2V, where y’ 
ig -the density, {2 the volume and V the velocity of the centre of 
mass of the body (the body being homogeneous) ; 
I= the impulse of the motion of the fluid. The time integral 
of f must be equal to the total impulse of the system, therefore: 
t 
framen Nad age es A. ORN) 
and 
dB dl 
=d. (5) 
dt dt 
When W is the “fluid resistance’, we have 
=— W Sees Sa! Go ek ee (ON 
i (6) 
and 
dl ; 
Wisent als i nd | ell ig ker 7 
de ER (7) 
1) See Kervin, Math. and Phys. Papers IV, p. 13 et seq. (1869). 
5) See H. Lams, Hydrodynamics p. 209. The formula has been proved there 
for a vortex system of finite dimensions; the integral, however, remains ety 
for an infinite system, when only condition (Ll) is satisfied. 
