The vane of a weathercock could thus be pivoted so as to point at any assigned angle with the wind; pivoted at the centre F the vane would set itself across the wind.

The movement of L towards the leading edge A' shows why a flat plate if free tends to set itself broadside to the relative stream as a position of stable equilibrium, seen realised in a falling card or leaf; and it explains the instability of the axial motion of an elongated body like a ship.

The stability of the course of a ship is secured only by a constant attention to the helm; but in a flying machine, by increase of Aspect Ratio, making the spread of the wings much greater than the axial depth, the vertical rudder requires little attention, but the vertical stability of the course requires incessant control by the horizontal rudder.

The curve LP for moderate value of a is seen running for some distance outside the curve LP'; and this shows how on Kirchhoff's treatment the lift may be obtained with a wing area. much smaller than was credited on the ancient theory, although a smaller pressure is assigned for normal impact.

For instance, with an average n = 6, and at 45 m/h,

for a small angle of 1 in n; and with n = 6, Q = 66,

This is very much

machine; Blériot takes

400
700

ft2/lb, the Demoiselle ft2/lb.

greater than is required in an actual
115
400

But an inspection of the figure shows that at small value of a the lift given on the Kirchhoff theory is much greater, requiring smaller wing area for given lift,

still much greater, nearly double, than is required in practice for wing area per lb of lift.

The discrepancy is attributed to the gain in efficiency due to camber of the wing; and a practical formula in general use is

(59)

Lift 1
Area n 8

S\2

(§)2, lb/ft3,

where is the slope of the chord of the camber.

We have shown that the counter current is insensible at the leading edge A' of attack; and as 1 in n is understood as the slope of the chord of the camber, the slope at the rear edge A is about 2 in n, which reduces the ft2/lb of wing area to a close agreement with practice.

But, as stated in the first clause of Report 19, there is no exact analytical theory at present for the calculation of a stream past a cambered wing, unless of two planes bent at an angle, and here the complication becomes almost intractable for practical use.

A liberal estimate of petrol and lubricating oil consumed is at 1 lb/H.P. hour, or one gallon per 10 H.P. hour: thus in a flight of 100 miles in two hours and a half at 40 m/h with a 50 H.P. Gnome engine, the quantity required would be

12.5
2

50 × 21 = 125 lb, bulking = 18.75, say 20 gallons,

With given lift, n varies as Q2, making the

and the hours of a journey varying inversely as Q, the H.P. hours vary inversely as Q2; so that about half the petrol is required to be carried if the speed over the journey is increased from 40 to 60.

On the ancient theory, n2 varies as Q2, making the H.P./lb constant, and the H.P. hours of a journey, and the petrol to be carried, inversely as the speed, or two-thirds for an increase of speed from 40 to 60 m/h.

This calculation ignores friction and head resistance, but it indicates in a general way the advantage and economy of high speed in flight.

Stability too is improved by speed, as well as economy in fuel.

LECTURE II

CALCULATION OF THRUST AND CENTRE OF PRESSURE OF

AN AEROPLANE

THE first lecture was confined to generalities, and mathematical detail required for a complete study of the subject, was avoided as much as possible.

But now we begin with a description of the Schwarz-Christoffel method of Conformal Representation, or Mapping, which shows us how to re-invent the original Helmholz-Kirchhoff solutions, and to extend them with ease and certainty to a large number of similar problems of greater generality, for which Report 19 may be consulted, on the Stream Line past a Plane Barrier, 1910.

The notation required has been given already in Lecture I., and we now proceed to state the theorem of Conformal Representation in the essential form as required for the subsequent application.

A point P, whose position is given by the vector z = x + yi, is to travel round a closed polygon or curve; and it is required to represent z as a function of some variable u, such that

(i.) u is real and diminishes from+∞ to as P performs a circuit of the polygon,

(ii.) points inside the polygon are to correspond to up + qi, but to upgi for points outside.

and the product of factors of the same form; and here u = a at a corner of the polygon, where the direction, or course, changes suddenly through a the exterior angle of the polygon.

The representation is called Conformal, because a small square on the z diagram corresponds to a small square on the u diagram, and the relative bearing of adjacent points is preserved for the two maps z and u, as they may be called; and there is thus no distortion although there is rotation and change of scale.

Conformal representation is then the same problem as mapping, and the first application on a large scale was the construction of a map on a given system of projection. Thus for instance the relation

connects the z stereographic projection of a hemisphere with the u Mercator chart. (Fig. 7.)

FIG. 7.

Suppose P travels round the polygon so that the inside is to the right hand ("starboard" a sailor would say), as in going round the clock; the angle a is positive when the change of course is to the right or starboard, due to porting the helm.

(3)

If the helm is put to starboard the angle a would be negative. Considering a single factor of (1), the relation

where M may be complex, and we put

(4)

M

= Neoi

= N (cos + i sin 0);