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Motion of a solid of

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the centre of reaction will be at its least. If, on the other hand, T-volution the line of the resultant impulse be at a greater distance than liquid. (A-B)ucos 22

from the centre of reaction, the angular motion AB will be always in one direction, but will increase and diminish periodically, and the centre of reaction will describe a sinuous curve on one side of that line; being at its greatest and least deviations when the angular velocity is greatest and least. At the same points the curvature of the path will be greatest and least respectively, and the linear velocity of the describing point will be least and greatest.

323. At any instant the component linear velocities along

& cos 0 and perpendicular to the axis of the solid will be and

A & sin

respectively, if o be its inclination to the line of reB

Ey sultant impulse; and the angular velocity will be if y be the

distance of the centre of reaction from that line. The whole
kinetic energy of the motion will be

& cos' 0 EsinA , &*y*



and the last term is what we have referred to above as the part due to rotation round the centre of reaction (defined in $ 321). To stop the whole motion at any instant, a simple impulse equal and opposite to & in the fixed “line of resultant impulse" will suffice (or an equal and parallel impulse in any line through the body, with the proper impulsive couple, according to the principle already referred to).

324. From Lagrange's equations applied as above to the case of a solid of revolution moving through a liquid, the couple which must be kept applied to it to prevent it from turning is immediately found to be

uv (A - B),

if u and v be the component velocities along and perpendicular Motion of to the axis, or [$ 321 (f)]

(A - B) sin 20 ૬


a solid of revolution through a liquid.

if, as before, & be the generating impulse, and @ the angle between its line and the axis. The direction of this couple must be such as to prevent from diminishing or from increasing, according as A or B is the greater. The former will clearly be the case of a flat disc, or oblate spheroid ; the latter that of an elongated, or oval-shaped body. The actual values of A and B we shall learn how to calculate (hydrodynamics) for several cases, including a body bounded by two spherical surfaces cutting one another at any angle a submultiple of two right angles; two complete spheres rigidly connected; and an oblate or a prolate spheroid.

325. The tendency of a body to turn its flat side, or its Observed

phenomena. length (as the case may be), across the direction of its motion through a liquid, to which the accelerations and retardations of rotatory motion described in § 322 are due, and of which we have now obtained the statical measure, is a remarkable illustration of the statement of $ 319; and is closely connected with the dynamical explanation of many curious observations well known in practical mechanics, among which may be mentioned :

(1) That the course of a symmetrical square-rigged ship sailing in the direction of the wind with rudder amidships is unstable, and can only be kept by manipulating the rudder to check infinitesimal deviations ;-and that a child's toy-boat, whether “square-rigged” or “fore-and-aft rigged *,” cannot be

+ "Fore-and-aft” rig is any rig in which (as in “cutters” and “schooners ") the chief sails come into the plane of mast or masts and keel, by the action of the wind upon the sails when the vessel's head is to wind.

This position of the sails is unstable when the wind is right astern. Accordingly, in “wearing” a fore-and-aft rigged vessel (that is to say turning her round ste to wind, from sailing with the wind on one side to sailing with the wind on the other side) the mainsail must be hauled in as closely as may be towards the middle position before the wind is allowed to get on the other side of the sail from that on which it had been pressing, so that when the wind

Applications got to sail permanently before the wind by any permanent addynamics justment of rudder and sails, and that (without a wind vane, or

a weighted tiller, acting on the rudder to do the part of steersman) it always, after running a few yards before the wind, turns round till nearly in a direction perpendicular to the wind (either “gibing" first, or “luffing" without gibing if it is a cutter or schooner)

(2) That the towing rope of a canal boat, when the rudder is left straight, takes a position in a vertical plane cutting the axis before its middle point :

(3) That a boat sculled rapidly across the direction of the wind, always (unless it is extraordinarily unsymmetrical in its draught of water, and in the amounts of surface exposed to the wind, towards its two ends) requires the weather oar to be worked hardest to prevent it from running up on the wind, and that for the same reason a sailing vessel generally "carries a weather helm*” or “gripes;” and that still more does so a steamer with sail even if only in the forward half of her length-griping so badly with any after canvasst that it is often impossible to steer :

(4) That in a heavy gale it is exceedingly difficult, and often found impossible, to get a ship out of “the trough of the sea,” and that it cannot be done at all without rapid motion ahead, whether by steam or sails :

(5) That in a smooth sea with moderate wind blowing parallel to the shore, a sailing vessel heading towards the shore with not enough of sail set can only be saved from creeping ashore by setting more sail, and sailing rapidly towards the shore, or the danger that is to be avoided, so as to allow her to be steered away from it. The risk of going ashore in fulfilment

does get on the other side, and when therefore the sail dashes across through the mid-ship position to the other side, carrying massive boom and gaff with it, the range of this sudden motion, which is called "gibing,” shall be as small as may be.

* The weather side of any object is the side of it towards the wind. A ship is said to carry a weather helm” when it is necessary to hold the “helm” or “tiller" permanently on the weather side of its middle position (by which the rudder is held towards the lee side) to keep the ship on her course.

+ Hence mizen masts are altogether condemned in modern war-ships by many competent nautical authorities.

under way


of Lagrange's equations is a frequent incident of “getting

while lifting anchor, or even after slipping from moorings :

(6) That an elongated rifle-bullet requires rapid rotation and gun. about its axis to keep its point foremost.

(7) The curious motions of a flat disc, oyster-shell, or the like, when dropped obliquely into water, resemble, no doubt, to some extent those described in § 322. But it must be remembered that the real circumstances differ greatly, because of fluid friction, from those of the abstract problem, of which we take leave for the present.


326. Maupertuis' celebrated principle of Least Action has Least been, even up to the present time, regarded rather as a curious and somewhat perplexing property of motion, than as a useful guide in kinetic investigations. We are strongly impressed with the conviction that a much more profound importance will be attached to it, not only in abstract dynamics, but in the theory of the several branches of physical science now beginning to receive dynamic explanations. As an extension of it, Sir W. R. Hamilton* has evolved his method of Varying Action, which undoubtedly must become a most valuable aid in future generalizations.

What is meant by “Action” in these expressions is, unfor- Action. tunately, something very different from the Actio Agentis defined by Newtont, and, it must be admitted, is a much less judiciously chosen word. Taking it, however, as we find it, Time aver. now universally used by writers on dynamics, we define the energy. Action of a Moving System as proportional to the average kinetic energy, which the system has possessed during the time from any convenient epoch of reckoning, multiplied by the time. According to the unit generally adopted, the action of a system which has not varied in its kinetic energy, is twice the amount of the energy multiplied by the time from the epoch. Or if the energy has been sometimes greater and sometimes less,

* Phil. Trans. 1834-1835.
+ Which, however (263), we have translated “activity" to avoid confusion.


Time aver. the action at time t is the double of what we may call the

time-integral of the energy, that is to say, it is what is denoted in the integral calculus by

age or energy doubled.

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where T denotes the kinetic energy at any time t, between the epoch and t.

Let m be the mass, and v the velocity at time t, of any one of the material points of which the system is composed. We have T= mv".

(1), and therefore, if A denote the action at time t,

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This may be put otherwise by taking ds to denote the space described by a particle in time dt, so that vdt = ds, and therefore A = Emvds ....

.(3), or, if x, y, z be the rectangular co-ordinates of m at any time,

A = | Em (¿dx + jdy + zdz)....... ...(). Hence we might, as many writers in fact have virtually done,

define action thus : The action of a system is equal to the sum of the arerage momentums for the spaces described by the particles from any era each multiplied by the length of its path.

Space average of momentums.

Least action,

327. The principle of Least Action is this :-Of all the different sets of paths along which a conservative system may be guided to move from one configuration to another, with the sum of its potential and kinetic energies equal to a given constant, that one for which the action is the least is such that the system will require only to be started with the proper velocities, to move along it unguided. Consider the Problem :Given the whole initial kinetic energy; find the initial velocities through one given configuration, which shall send the system

unguided to another specified configuration. This problem is Turgut too essentially determinate, but generally has multiple solutions

(8 363 below); (or only imaginary solutions.)

General Problem of aim.

High aim and low aim for same target.

fur off to be reached.

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