This last operation for the determination of L is due to Lord Rayleigh; although not so very formidable after all, it must have appeared so to the inventor working out these integrations for the first time with no knowledge of the result.

The three integrations, of thrust T, length x, and centre of pressure L, are typical of what is required in the other extensions of this Kirchhoff problem, which follow in Report 19.

A change of into n will give the fluid motion past the barrier AA' when it is bent at B to an angle ; and then

A BA' may replace a cambered wing, touching at A and A'. But now the integrations for A'B, BA become intractable, and so also for the thrust T, because

The lift of an aeroplane AA' arises from the gauge pressure on the under face, due to the defect in velocity q, and the opening out of the stream lines in Fig. 12.

The pressure in rear of the plane in its wake is taken as atmospheric, up to and along the skin of the bounding stream AJ and A'J', the air behind being at rest relatively to AA', or in a state of vortical turbulence at average atmospheric pressure.

Any thrust on the plane AA' must be due to an excess of pressure over the atmosphere, gauge pressure so called, not absolute, on the under attacking face; this gauge pressure arises from a diminution of velocity of q below Q, and diminution of Q2 q2

dynamic head, from to and an equivalent rise of static 2g 2g

pressure head, due to the broadening of the interval between the stream lines close to AA', especially in the neighbourhood of B, the branch point.

This shows in a general way why L, the C.P. (centre of pressure) is away from F, the C.F. (centre of figure) and more towards B, so that the tendency of the fluid reaction is to turn the plane more broadside to the stream.

A popular figure of the stream lines past a cambered wing, as here in Fig. 13, showing no such broadening, would imply at once to our eye an absence of all thrust and lift; the figure should be more like Fig. 14.

In the electrical law of flow where there is no wake, as here in Colonel Hippisley's diagram (Fig. 15), the broadening of the stream is shown near a branch point; but as the stream lines close in behind symmetrically, there is no resultant thrust on the cylinder by the stream tending to move it, but a couple only tending to turn the cylinder broadside to the current, arising from excess of pressure near the two opposite branch points.

The magnitude of the couple on an elliptic cylinder in found to Q2

be sin 2a times the weight of the column of liquid with a

2g

The couple is shown experimentally by letting a card drop in the air from the horizontal or vertical position, or by projecting it with a spin, observing how the card turns gradually into a vertical plane.

So, too, the stability is assured of the forward course of the flying machine by making the spread of the wings three or four times the depth from front to back. It is supposed, too, that the stability can be improved by giving the wings a dihedral angle, as at the top of the stroke of the wings of a bird.

The existence of a couple on a flat or elongated body moving in a medium, air or water, tending to set it broadside to the motion is discussed in Thomson and Tait's" Natural Philosophy," and in the Hydrodynamical Treatises of Basset and Lamb, in a treatment rather abstruse.

But an explanation can be given in the manner called "elementary," depending on the principle of momentum.

Take a body like an elliptic cylinder, as in Colonel Hippisley's diagram (Fig. 16), and supposing it moving with velocity Q, and velocity components U and V, being held so as to prevent rotation about the axis of the cylinder.

If there is no surrounding medium, the components of momentum, W and W sec-lb of the body weighing

U
g

V 9

W lb will remain unaltered while the centre moves from 0 to O', so that the vector OH of momentum will move to O'H'' in the same straight line 00', and there is no change of momentum and no force or constraint is required.

X

FIG. 16.

But if the body is surrounded by a medium, of which it displaces W' lb, the velocity components U and V will give rise to

momentum-components in the medium, c1W'

U

g'

V

C2 W" where

c1 and c2 are different constant numbers depending on the shape of the body; and as the broadside motion will give the greater momentum, c2c1.

The body must now be held to prevent rotation; because at O' the momentum of the medium has changed from OF to O'F", not in the same straight line 00'; and the change of momentum is the impulse couple N1 OF. OD, acting on the medium, against the clock, OD being the perpendicular on O'F".

=

The medium reacts on the body with an equal and opposite impulse couple N1, tending to set it broadside; and if t seconds is the time from 0 to O', the impulse couple given by its components against the clock is

the accumulated effect in a time t of an incessant couple

and this is the expression found in Hydrodynamics.

For an elliptic cylinder, as in the diagram (Fig. 17), it is found by theory that

and for a length ft in a medium of density w lb/ft3, W' = пwabl,

where W" is the weight of medium displaced by a cylinder of cross section the circle on the diameter SS' joining the foci S and S'.

This reduces for a card of breadth 2a, with b = 0, to

(60)

=

N πωα21

=

UV Q2
Aan sin a cos a,
g
Cg

assuming the electrical law of flow round the edge A and A'. But this would be qualified by the factor

3332

(1 + sin a)2

in Kirchoff's discontinuous motion, where the couple, in accordance with (53), would be

The calculation is not simple of c1 and c2 for other figures, such as a solid of revolution; but it is required in the discussion of the stability of an air ship or submarine boat, in its effect on steering and loss of metacentric height.