May 9, 1912] 
NATURE 
253 
of time. The area which, when loaded with 150 Ib., 
drops at the rate of 1 foot per second, is ae 
or 10,600 square feet, that is, a circle of 113 feet 
diameter. 
With such a parachute a man could by climbing 
keep himself stationary in the air. 
It is not necessary, in order to impart this 
momentum to the air, that the surface should itself 
have this area of 10,600 square feet. The same 
momentum may be given by a much smaller inclined 
surface moving horizontally. 
If a perfectly efficient screw or inclined plane were 
a physical possibility, there would be nothing to pre- 
vent people from flying by their own muscular effort, 
and it is worth while to examine the causes which 
prevent the realisation of such a result. 
Fic. 2 
We will now consider more closely the causes which 
produce the very marked difference between the 
theoretical curves given in Fig. 2 and the correspond- 
ing quantities as determined by experiment. _ 
It is well known that the fluid with which mathe- 
maticians deal, and which is supposed to surround 
the plane in Fig. 2, is an ideal body which is with- 
out viscosity (that is, opposes no resistance to shear), 
and that in contact with a solid it experiences no 
frictional retardation. 
In such a fluid pressure and velocity are connected 
by an invariable law, the sum of the potential and 
kinetic energies of any portion of the fluid re- 
maining constant for all time. 
This law, together with the necessary con- 
dition of continuity, which for an incompressible 
fluid merely implies that the volume of a given 
mass of fluid remains constant, no matter what 
shape it takes, constitutes the foundation of all 
the propositions regarding the stream lines of 
a perfect fluid which have hitherto been worked 
out, and for such a fluid the stream lines indi- 
cated in Fig. 2 are an exact solution of the 
problem. 
Now real fluids differ from the perfect fluid in 
having both viscosity and surface friction. They 
require that work should be done if distortion ‘is 
going on, and they adhere to the surfaces of solids 
immersed in them. Thus a plane which, if moving 
edgewise in a perfect fluid, would meet with no 
resistance, does meet with resistance in a real fluid 
on account of the adherence of the fluid to the solid 
surface and the consequent distortion produced in the 
neighbouring layers of the fluid. 
It is true that for fluids such as water and air the 
viscosity is so small that the direct effects would 
hardly be noticeable. Indirectly, however, they have 
immense influence, and it is not too much to say 
that the most remarkable features in the flow of the 
winds, tides, and streams are due to the modification 
of stream-line motion set up by fluid friction and 
viscosity. 
NO. 2219, VOL. 89] 
The indirect action referred to depends on the fact 
that when a stream is retarded by friction the velocity 
is reduced, although the pressure remains unchanged, 
and thus the fundamental relation which connects 
velocity and pressure in a jerfect fluid is violated. 
So long as the stream concerned is of constant section 
and is neither accelerating nor retarding, as, for 
instance, when the flow is through a straight pipe 
of uniform bore, the effect of friction shows itself 
merely by rendering the stream lines irregularly 
sinuous, in a way which has not yet been investi- 
gated, and as giving rise to a resistance which is 
proportional to a power of the velocity something 
rather less than the square, i.e. to the 185th or roth 
power. 
When, however, the stream is divergent (so that 
in the absence of friction the velocities and pressures, 
although constant across each section, change from 
one section to another, but keep the total energy of 
the flow across each section the same), the effect of 
friction and viscosity is much more conspicuous. 
On the up-stream side of the plane friction does 
little to modify the conditions except in the neigh- 
bourhood of the edges, but down stream we find, 
instead of a pond of still fluid, a complex wake con- 
sisting of a central current moving forwards towards 
the plane, bordered by a series of eddies the origin 
of which is of the same nature as those just referred 
to in the expanding channel, namely, to degradation 
of the streams passing round the edges of the plane, 
which, having insufficient velocity to follow the 
stream-line path of Fig. 2, are deflected inwards and 
become involved with the reversed central stream, 
about half the fluid in each eddy being supplied from 
up stream and half from the wake. 
The eddies are formed periodically, growing to a 
certain size, and then, breaking away from their 
place of birth, they form part of the train which 
borders the wake current. The wake current itself 
is due to the constant removal of fluid in this way 
from the back of the plane, and the fact that the 
outflow from the back has its maximum velocity 
close to the edge where the composite eddy is being 
formed shows that the pressure on the back of the 
Fic. 3.-—Frictional flow : stream oblique to plate. 
plane is lower at the edges than in the centre. Hence 
it could be stated with certainty, even’ without any 
experiment, that the total resistance of a plane must 
be greater than pv?“ =!" * _, 
4+7 sina 
pressure over the rear surface is uniform and equal 
to the general pressure at a distance. 
Experiment, however, is required to determine the 
actual resistance, and when the plane is broadside to 
the stream this is found to be about half as much 
again as the head resistance alone, or about 20 or 
which assumes that the 
| 25 per cent. greater than the dynamic head x the area 
of the plane. 
When the angle a is small, as it always is in flight, 
the character of the wake takes the form shown in 
Fig. 3. Here the wake stream is only recognisable 
| as a reversed current quite close to the plane, and 
the small eddies as fast as they are formed are so 
