4 
particle at P in Fig. 1, we could trace its path relatively to 
the cylinder by superposing on its actual motion a backward 
velocity w parallel to Oz. It can be shown that this relative 
path is a curve PBC whose equation in terms of y and 7 is 
(1 ==) =, « : 3 : d (1) 
where 6 is the distance of the particle from the axis Ox when 
at an infinite distance before or behind the cylinder. These 
curves, given by different values of 6, would be the actual 
paths of the fluid particles if the cylinder were at rest and 
the fluid were streaming past it. In the case under con- 
sideration the cylinder is moving and the fluid is at rest at 
infinity; hence the actual path of a particle may be imagined 
as the path in (1) referred to axes moving with uniform 
velocity w. ‘The equation of the path was first obtained by 
Rankine! in the form of a relation between the ordinate y 
and the radius of curvature p. It can be deduced from 
Fig. 1. 
We have pd(26)ldt = velocity of particle at P = ua?/r?, 
By writing down the velocity of P relative to O in a direc- 
tion at right angles to OP we have 
° 
2 Sin 6. 
o =u sin 6 + 
From these two equations, with y for rsin6, we obtain 
2py(1 + a?/r?)=a2. But relatively to the cylinder the 
particle lies on the curve given by equation (1) above; hence, 
substituting for a?/r? we find the result 
5 =a — 3) Er eT restn cee Wis, Hos 
As Rankine pointed out, this represents in general a case of 
the ‘elastic curve’; and, in fact, the path of a particle is one 
loop of a coiled elastica. We can complete now the solution 
of (2). For any given particle, fixed by the value of b, if 6 
'W. J. M. Rankine, Phil. Trans. A., vol. 154, p. 369, 1864. The 
result is erroneously attributed to Maxwell in the article on hydremechanics 
in the Encyclopedia Britannica, 11th ed. 
83 
