In artillery the theory is useful for determining on gyroscopic principles the spin requisite for stability of an elongated shot fired from a rifled gun.

When the external shape is assimilated to a prolate spheroid,. generated by the revolution of an ellipse of axes 2a, 2b (Fig. 17), and when the length in diameters 2a/2b is denoted by x, we find by hydrodynamical theory, on the electrical law of flow,

where A and B are determined from the equations,

For a sphere, x = 1, A = B = 33, C1 = C2 , and the effective inertia of a sphere is increased by half the weight of the medium displaced.

But for broadside motion of a cylinder, x = ∞, A = 0, B = 1, c2 = 1; so that the inertia is increased by the weight of the medium displaced.

Thus a spherical balloon or cylindrical air ship of weight W lb,

displacing W' lb of air, will start to rise with acceleration

(64)

W W
WW 9, or

W' W
W+W 9.

A spherical bubble of air in water, where W is insensible compared with W', will start up with acceleration 2g.

A flexible sheet in tension t, separating the current of density w from a dead wake of different density w', as with water past air, and flapping like a flag or shivering as a sail, would swing in waves of length à advancing with velocity U, such that in accordance with wave motion theory

so that the waves break up into vortex motion of

The practical question arises as to the advantage of a short length of fringe or flap at the edge A of the aeroplane, so as to retard the formation of a vortex, such as seen at the lee of the chimney top. Some such arrangement can be seen in the new flying machine.

In the absence of these wings the equipotential lines of the electric field would bend round the barrier symmetrically (as in Fig. 18), and IB would be the branch of a hyperbola with foci at A and A', continued on the other branch B' I' at zero potential. This electric field would represent the analogous streaming motion, realised by coloured filaments in Hele Shaw's experiments, in a viscous liquid, where the motion is slow and no perceptible eddy is formed at A and A'.

The liquid is then said to stream on the electrical law of flow; and the more general case where AA' opens out into a confocal ellipse is shown in Colonel Hippisley's diagram here (Fig. 15),

which can be interpreted electrically as representing the disturbance in a uniform field by the presence of an elliptic cylinder to earth.

A magnetic interpretation can also be given to his diagram as a generalisation of Maxwell's Figure XV., of a magnetised cylinder in a uniform magnetic field.

It is gratifying and of great assistance, too, when the mathematical analysis of a mechanical question can be made to serve in another interpretation; and so the student of electricity should be interested in following up the electrical analogue of this hydrodynamical problem.

We must suppose the uniform horizontal current to represent a uniform vertical electrical field, and this field to be disturbed by the uninsulated plane strip AA'; while the skin stream lines AJ, A'J' must be replaced by wings of flexible gold or tin foil, as in the electrometer, but stretching away to infinity.

The wake of the plane AA', which to the pilot seems stationary behind him, is being dragged along through the air by the flying machine.

Any machine following which flies into this wake or backwash will seem as if it is entering still air; the lift is lost and the machine will drop. This may happen if the pilot is following at a different level, above or below, so that in passing another machine it is prudent for the pilot to take a course to one side.

LECTURE III

HELMHOLTZ-KIRCHHOF THEORY OF A DISCONTINUOUS STREAM LINE

THE normal conditions of the aeroplane of a machine flying high up in the air are represented on the Kirchhoff theory by the diagram 4 and 10, representing the state of flow relative to the pilot; the analytical conditions have been discussed in Lecture I. and II., and will be found in § 6 of Report 19.

But there is an advantage in extending the theory to a more general case, and beginning as in Report 19, § 2, with a plane barrier oblique to a stream of finite breadth (Fig. 19), as the extension is not essentially more complicated, and it throws light on the simple case of an infinite stream.

No change is required in the £ diagram (Fig. 19), except that i, j, j', on aa' are now distinct; but now in the w diagram the outside stream lines come from infinity into view, and the diagram has the advantage of being closed, as in Euclid I. 27, so that the difficulties at infinity are kept under observation.

In the w diagram there are three stream lines in the boundary, which we denote by mi, m2, m3; and my is the outer stream line from I to J; y = m2 the inner skin from J to A, from A along AA' to the branch point B and then to A', and again along A'J'; = mg is the other outer stream line from J' back

to I.

There is also the curved branch IB of the m2 stream line along which the current divides; but as IB lies in the interior of the fluid, and the velocity q and its direction both vary along IB, the corresponding part ib in the w and 2 diagram must be excluded from the boundary; ib is straight in the w diagram, but curved in the £ diagram, from b at midway height of a and a', to i at height a.

The branch point B where the velocity is zero requires

= 0 at u = b, so that, by II. (1),

аф

du

m2

when resolved into partial fractions, with the factor appropriate for making w change suddenly by m1 - m2, M2 — M3, M3 — M1, as u passes through j, j', i, in accordance with II. (15).

mı

M.F.

E