the North-East and South-West. Let ABC denote the position of the sail, and EB the direction of the wind. Suppose the velocity of the wind resolved into two components, one along AB and the other along DB at right angles to AB; the latter only is effective in producing a pressure on the sail : see Art. 473. Thus we get a force acting on the sail in the direction DB; this force is not in the direction in which

B

M

A

the ship is to move, but we may suppose it resolved into two components, one acting from the stern to the bow and the other at right angles to this. The former component urges the ship in the required direction; the latter would tend to urge the ship sideways, but it produces little effect because of the resistance opposed by the water to the large surface which the ship presents sideways; what effect it does produce must be counteracted from time to time by the action of the rudder.

580. It will be seen that two or three sails may be used parallel to each other; thus in the diagram a second sail, as large as the first, might be placed near the stern, so that neither of them should intercept the wind from the other. We have supposed that the angle between the direction of the wind and that of the course is a right angle; but this is not essential: the precise position in which the sails must be put in order to secure the greatest velocity will depend on the angle between the directions of the wind and the course, and can be determined in every case by trial. A ship cannot sail directly against the wind, but it may approach very nearly to such a course. For instance the wind might be from the North East, and yet the ship sail from South to North. If the wind is directly, or nearly directly, opposed to the desired course the ship must adopt a zigzag course. Thus, for instance, if the ship cannot sail directly from South to North it may sail towards the North East for some dis

tance, and then change its direction and sail for some distance towards the North West; thus by one or more such tacks the ship may reach its proposed port.

581. We proceed to explain the use of the Rudder. Suppose the ship sailing from South to North, and let the rudder be in its middle position, that is let the rudder and the keel be in one plane. Then the resistance of the water acts on one side of the ship precisely in the same manner as on the other, and so does not tend to change the direction of the motion. Suppose then that in order to avoid an obstacle, as for instance another ship, it is desirable to change the direction of the ship by turning the bow towards the right hand side: to effect this the rudder is turned towards the right hand side. Let RT denote the position of the rudder thus turned to the right-hand side by reason of the passage of the rudder through the water, with the ship, a resistance is exerted on the rudder in the direction MN which is at right angles to RT. One effect of this would be to push the whole ship in the direction MN; but a more important effect is to give the ship a rotation in the horizontal plane round its centre of gravity; see Arts. 344 and 345. Thus in virtue of this rotation the bow of the ship turns towards the right hand, and the stern towards the left hand. When the proper change of direction has been produced the rudder is put back again into its middle position.

;

582. Rowing Boat. In Art. 198 we just alluded to the oar of a boat as affording an example of a Lever of the second class; but it will be instructive to consider the forces which act on the boat. The man who rows exerts by his hands a certain force on the oar or the pair of oars which he grasps: we will suppose that he holds a single oar, and we will denote the force he exerts on it by P. In consequence of this effort of the man a certain force is exerted by the oar on the row-lock and by the row-lock on the oar; this we will denote by Q. If the man holds two oars P is the sum of the forces he exerts on the oars, and the sum of the forces at the row-locks. Now Qis greater than P by the nature of the lever, for the oar may be considered to form a lever of the second class with the fulcrum at the blade in the water. It might

at first sight seem that Q is the force which the man applies to the boat to maintain it in motion. But we must remember that he cannot bring his strength to bear on the oar unless he pushes with his feet against the boat in the opposite direction. The force thus exerted is practically equivalent to the force which he exerts by his hands, that is to P. Hence he really exerts on the boat the force Q-P. It is only through having the external water as a fulcrum that he is able to bring the force Q which is greater than P to act in the contrary direction, and thus to leave Q-P to be effective on the boat. Children in a railway carriage may sometimes be seen pushing at the front of the carriage in order to start it; they do not know that, since action and reaction are equal, the force of their hands is balanced by that of their feet in the contrary direction.

583. The preceding Article suggests the remark that in a Mechanical Problem there may be more than one distinct body involved, and that in order to discuss the equilibrium or the motion we may have to consider each body separately. Thus if a man row a boat with a pair of oars there will be four bodies, namely the boat, the man, and the two oars, each acted on by its special system of forces. We will illustrate the matter by considering the case of a wheelbarrow which contains a load, as a stone, and is held by a man in the position just previous to motion so that the whole is in equilibrium. The stone is acted on by its own weight downwards, and by the resistance of the wheelbarrow at the point or points of contact. The wheelbarrow consists of two distinct parts; namely, the trough-part including the legs and arms, and the wheel-part consisting of the wheel with its axle. The trough-part is acted on by its own weight, by the pressure of the stone, by the action of the wheel-part at the ends of the axle, and by the force exerted by the man's hands. The wheel-part is acted on by its own weight, by the action of the trough-part at the ends of the axle, by the resistance of the ground in a vertical direction at the points where the wheel touches the ground, and by the friction of the ground at the same point in a horizontal direction. The man is acted on by his own weight, by the pressure of the

wheelbarrow on his hands, by the resistance of the ground in a vertical direction, and by the friction of the ground in a horizontal direction. By the law of the equality of action and reaction the pressure of the stone on the wheelbarrow is equal and opposite to that of the wheelbarrow on the stone; the action of the trough-part on the wheelpart is equal and opposite to that of the wheel-part on the trough-part, and the action of the man on the wheelbarrow is equal and opposite to that of the wheelbarrow on the man. We suppose the ground to exert a friction, which it will do in practice; but it is theoretically conceivable that there is no friction. In this case as the man's weight and the action of the ground are both vertical, so must the action of the wheelbarrow on him also be, in order that he may be in equilibrium; that is, he must pull the wheelbarrow in a vertical direction upwards and be pulled by it in a vertical direction downwards. This would require him to lean backwards so as to throw his centre of gravity behind the points at which he presses the ground; for the upward force on him must fall between the two downward forces that he may be in equilibrium. In this case all the other forces which act on the trough-part being vertical the action from the wheel-part must also be so; then the other forces on the wheel-part being vertical there can be no friction on it.

LII. WORK. ·

584. In modern treatises on Practical Mechanics the term Work is employed in a peculiar sense; and various useful facts and rules are conveniently stated by the aid of the term in this sense. We propose accordingly to give some explanations and illustrations which will enable the reader to understand and apply such facts and rules.

585. The labour of men and animals and the power furnished by nature in wind, water, and steam, are employed in performing operations of various kinds, such as drawing loads, raising weights, pumping water, sawing wood, and driving nails. In these, and similar operations, we may perceive one common quality which is adopted

as characteristic of Work, and suggests the following definition; Work is the production of motion against resistance.

586. This definition will not be fully appreciated at once; the beginner may be inclined to think that it will scarcely include every thing to which the term work is popularly applied: he will find however as he proceeds that the definition is wide enough for practical purposes. According to this definition a man who merely supports a load does not work; for here there is resistance without motion. Also while a free body moves uniformly no work is performed; for here there is motion without resistance.

587. Work, like every other measurable thing, is measured by a unit of its own kind which we may choose at pleasure. The unit of work adopted in England is the work which is sufficient to overcome the resistance of a force of one pound through the space of one foot or we may say practically that the unit of work is the work done in raising a pound weight vertically through one foot.

588. The term foot-pound is used in some books instead of the term unit-of-work: so that foot-pound may be considered as an abbreviation for one pound weight raised vertically through one foot.

589. Some English writers prefer to use the French system of weights and measures instead of our own; this system is explained in the Mensuration for Beginners. In this system the standard weight is the kilogramme which corresponds to about 15432 English grains, that is rather more than two pounds Avoirdupois; the standard length is the metre which is about 39 371 English inches. The unit of work is that done in raising one kilogramme through a vertical height of one metre; it is called a kilogrammetre.

590. The term horse-power is used in measuring the performance of steam engines. Boulton and Watt estimated that a horse could raise 33000 pounds vertically through one foot in one minute; this estimate is probably too high, on the average, but it is still retained, so that a horse power means a power which can perform 33000 units of work in a minute.