A second lecture, of one minute, would carry on with the Statics of the alteration of trim of the floor on the springs, due to passengers entering or leaving, and as the carriage is accelerated or retarded.

The Law of the Spring is assumed as an experimental fact, based on Hooke's vague statement of the law—

Ut tensio sic vis.

A third lecture could be devoted to the simple pendulum, and the length required to beat time with the oscillation of a carriage body on the springs, vertical, pitching and rolling.

Thus the vertical oscillation should synchronize with a pendulum of length equal to the set of the springs, the vertical distance the carriage body sinks down on superposition; the law of the spring being supposed to hold.

This is verified with a spring balance and a weight, a 32-lb shot, provided the scale can be graduated uniformly.

But the logical and simple statement of the formula for the beat of the pendulum is

if L denotes the pendulum length which beats the second; as it is L which is determined experimentally, and g is derived ༡ from it by the relation

The rolling and pitching oscillation of the carriage body on the springs would introduce the idea of angular inertia, with which this lecture began; this is shown in its simplest form in the carriage wheels, in adding to the linear inertia of the carriage. The measurement of moment of inertia, or second moment, would run into a fourth lecture, provided we had the unlimited time. at the disposal of Marchis in his Sorbonne lectures.

LECTURE V

THE SCREW PROPELLER

THERE is no exact theory, it must be conceded, of universal acceptance for the screw propeller, and reliance is placed chiefly on an empirical factor based on experience and model experiment, employed in a formula which satisfies the condition of mechanical similitude, so as to predict from a small scale experiment the performance to be expected of the full size machine.

A rational theory can be given of a hydraulic machine or turbine, when the water is compelled to follow a definite path; but where the fluid, air or water, is free to take its own course, as in the screw propeller, no exact treatment is possible until the stream lines have been determined.

Where the screw works in open water or air, the stream line is free to take a line of least action, and the shape is influenced to a great extent by the hull and fixtures in the neighbourhood, and the relative position of the propeller, effects which cannot be considered in a single formula.

Numerous theories will be found in the Abstracts of the Report of the Aeronautical Committee due to various experimenters, and one initial difficulty is to reconcile the conflicting notation employed by each writer; it is time this notation was standardised.

But the formulas are found to be in general agreement in making the thrust T proportional to

3. The square of the blade tip velocity, U f/s;

4. The slips; or more accurately to the product s (1-8).

With an empirical factor f the formula for the thrust may then be written

and this is the weight of a cylindrical column of the fluid, of cross section S, and height

The formula agrees then in making T=0 when s=0 and there is no slip, and the screw, of uniform pitch, advances in the fluid as if in a solid nut; and also when s = 1, and there is no advance, and the screw cuts a hole in the fluid and swirls the fluid round.

But with a slip s between 0 and 1, the reaction of the fluid is against the rear of a blade, and a thrust is obtained.

A negative value of s would imply that the screw was being turned by the stream through it, as a windmill or turbine.

Working on the sails of a windmill or ship the wind strikes the rear of a sail and urges it forward, as in Fig. 48.

If OW' represents II', the true wind over the water, and OT the velocity of the ship through the water, then VW represents the apparent wind Q, as felt sweeping across the deck and filling the sail; and the direction of 'I' is given by the vane on a mast, or smoke from a chimney.

(3)

On the Newton theory the thrust on sail area A ft2 is given by

as we have taken previously when Q is given in f/s; but when the speed is given by K in knots of 100 ft/minute,

because 60 knots is 100 f/s, 12 knots 20 f/s.

FIG. 48.

(5)

On the diagram of velocity of Fig. 48, 'W'V'' = a, vw = Ow - Ov, Q sin a

=

W sin ẞV sin 0,

and the propulsive force in the line of the keel is

if II' and V are measured in knots.

One H.P. of 33,000 ft-lb / min. is 330 knot-pounds, so that

Working this out for a ship of the size of the Preussen, spreading A=40,000 ft2 of canvas, with 30°, a 60°, Q = 12 (18) knots, 0 = = V10 (15) knots, the sail H.P. is then about 450 (1,534).

An ice boat would run like a windmill unloaded; and at full speed OV', the vane on the mast would point parallel to the sail.

For the screw propeller, as far as theory can go at present, we begin with Rankine's treatment in the Transactions of the Institution of Naval Architects, 1865, following his notation as closely as possible.

A screw surface, of a true screw, is swept out by a straight line intersecting an axis at right angles; and the line advances along the axis and turns at the same time in a constant ratio; and the advance for a complete revolution is called the pitch, and denoted by p, and measured in feet.