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away through D to the condenser, so that the piston is forced down by the pressure of the steam above it. This engine is sometimes called the double-acting steam engine from the circumstance that the force of steam drives the piston alternately up and down.

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566. The High-pressure Steam Engine. struction is much the same as in Watt's steam engine, but there is no condenser. The steam has a pressure many times greater than that of the atmosphere, and instead of being condensed after each stroke it is permitted to escape into the open air. This is the form of steam engine used on railways.

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567. We have given only a brief sketch of the steam engine; there are many important details connected with the subject, for an account of which the student must consult special treatises. One of the most remarkable contrivances due to Watt is called the Parallel Motion. the atmospheric steam engine the ends of the lever are arched, and chains passing round them are connected with the ends of the rods which move up and down; thus the piston E can pull the end F down, but cannot push it up. Watt devised a system of jointed bars which allowed the piston rod to move vertically and F to describe an arc of a circle, while the piston rod could push as well as pull the end of the lever. The motion is very important not only in the steam engine but in various cases where motion in a right line is to be transformed, as it were, into motion in a circular arc, and the contrary; attention has recently been drawn to this transformation by some fine researches of Professor Sylvester in relation to a method invented by M. Peaucellier.

LI. FAMILIAR APPLICATIONS.

568. In this Chapter the principles which have been already explained will be applied to some familiar examples, in some cases taken from well-known toys of children.

569. The Kite is memorable as having been a favourite toy with Newton; and the younger Euler, a well-known

mathematician, has devoted to it a memoir in the Transactions of the Berlin Academy for 1756. It is unnecessary to describe an object so well known as the kite; we will suppose it floating in the air and at rest. There are three forces which act and maintain equilibrium; the weight of the kite, including the tail; the force of the wind; and the tension of the string. The weight acts vertically downwards. The wind may be taken to blow horizontally, but its force must be supposed to be resolved into two components, one along the surface of the kite, and the other at right angles to the surface; it is only the latter which produces any effect on the kite, for the former would be like a wind gliding over the surface of the kite and not pressing it see Art. 473. The tension of the string acts in the direction of the string at the point where it leaves the kite; but usually the string near the kite is, as it were, divided into two, one going to a point near the upper end of the kite, and the other to a point near the lower end: in this case the tensions. of the two strings are equivalent to the tension of a single string the direction of which is that of the kite-string at the point where it is divided into two. The three forces which thus act on the kite must fulfil the proper conditions in order to produce equilibrium; this will require that their directions should meet at a point, and that their magnitudes should be in the proper proportion.

570. The kite then adjusts itself to a suitable inclination, and the tail adjusts itself to a suitable position, so as to bring about the precise circumstances necessary for equilibrium; but it would not be easy to state in words exactly what these must be. If we consider the kite alone we can find the situation of its centre of gravity by the experimental method of Art. 170; but when the tail is attached the situation of the centre of gravity of the whole will depend on the position taken by the tail. The weight of the kite alone, or of the kite and the tail, can easily be ascertained. If we consider the kite alone, the points at which the force of the wind on it may be supposed to act can be found. For we may conceive the force of the wind to consist of parallel pressures on all the portions of the face of the kite, the pressures being equal on equal

portions of the face. The point where their resultant acts will be the centre of gravity of the face of the kite; this is not necessarily the centre of gravity of the whole kite, for there is usually a straight piece of wood running down the middle of the kite, and a curved piece of wood at the top; but it would be the centre of gravity of the kite if these pieces of wood were removed. We might cut out a figure in pasteboard, or thick paper, of the shape of the kite, and then its centre of gravity would be practically coincident with the centre of gravity of the face which we require. But when the tail is taken into consideration also, since there must be some action of the wind on that, it is impossible to say at what point the resultant force of the wind may be supposed to be exerted. The string itself will be in equilibrium when all the system is at rest, and this gives rise to an interesting problem though too difficult for an elementary book. The forces acting on the string are its own weight, the pressure of the wind, and the tensions at the two ends, where the string may be considered to be held fast. It is obviously seen by trial that the string does not take the form of a straight line.

571. The See-saw. A plank is put across a log of wood, and one boy sits on one end of the plank and another boy on the other end. The plank turns round the part in contact with the log as a fulcrum, and so the boys move alternately up and down. If the boys are of unequal weight their positions must be adjusted according to the principle of the lever; the distance of the heavier boy from the fulcrum must bear the same proportion to the distance of the lighter boy from the fulcrum as the weight of the lighter boy bears to the weight of the heavier boy. The motion is kept up by each boy in his descent touching the ground with his feet, which diminishes his pressure on the plank, and gives to the weight of the other boy a momentary superiority. Or the descending boy may push firmly against the ground, which tends still more to send him up and bring the other boy down.

572. The Swing. This well-known contrivance bears a resemblance to a pendulum. The person in the swing

may have his motion kept up by receiving an occasional push to counteract the effects of the resistance of the air and the friction at the points of support. He may also keep up, and even increase, his own motion by crouching in the swing when at the highest point and rising in the swing when at the lowest point. It is not possible to establish this result strictly in an elementary manner, but we may give some explanatory remarks. When the man crouches in the swing he puts his centre of gravity further from the fixed point than it was before; thus the result is like that of augmenting the length of a simple pendulum while starting it at the same inclination to the vertical: therefore the centre of gravity descends through more vertical space and so has a greater velocity at the bottom than before: see Art. 318. Again, when the man rises he brings his centre of gravity nearer to the points of support; thus the result is like that of diminishing the length of a simple pendulum while starting it with the same velocity from the lowest point; therefore the pendulum must move through a larger angle than before, in order that by passing through the same vertical space as before it may lose the velocity with which it started. Thus by either crouching at the highest point or rising at the lowest, the motion is increased; and of course if both changes are made the effect produced is all the greater.

573. The Top. A few words must be devoted to this striking toy; the tops introduced of late years, which continue spinning for several minutes, are especially interesting. The reader will perhaps be disappointed by the remark that it is impossible to give any satisfactory account of the subject in an elementary book; but such is the fact; for the discussion of the motion is really a most difficult problem, requiring the highest mathematical resources. One peculiarity of the motion is the steady character which sometimes belongs to it when the top in popular language is said to be sleeping. We know that it is almost impossible to balance the top on its point when the top does not rotate, and the question then is, how can the top be kept from falling when it rotates rapidly? Suppose that at any instant the centre of gra

vity of the top, instead of being vertically over the small base, is a little to the right-hand side; then before the top has time to fall over towards this side the rotation carries the centre of gravity round to the opposite side, and thus prevents the fall towards the right-hand side. Another peculiarity of the motion is the fact that under certain circumstances the top raises itself from an inclined position to a vertical position; this is done by the aid of the friction. It may suffice to convince the reader of the difficulty of the subject just to state that some reasoning given_by_Euler in explanation of this fact, and adopted by Dr Whewell, is pronounced by a well-qualified judge to be vague, inconclusive, and directly the reverse of the truth. A popular explanation has been given which assumes the existence of a friction acting vertically upwards at the point of contact with the ground; but in the first place friction acts horizontally and not vertically, and in the second place the effect of a vertical force would be to increase the inclination of the top to the vertical instead of diminishing it: see Art. 345.

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574. The Popgun. This is a well-known toy. pellet is put at each end of a hollow cylinder so as to keep the cylinder air-tight. One of the pellets is pushed forwards by a stick, and thus the air between the two pellets is compressed, and its elastic force increased. The other pellet then is pushed out as soon as the pressure of the compressed air behind is greater than that of the atmosphere in front, together with the friction between the pellet and the hollow cylinder.

575. The Squirt. A hollow cylinder is tapered off to a point where there is a hole; this end is put under the surface of water in a vessel, and a piston drawn nearly through the cylinder just as in the common pump: water is forced in by the pressure of the atmosphere. The water will not flow out of itself if the squirt is removed from the vessel, but may be expelled to some distance by driving the piston back rapidly.

576. The Sucker. A string is fastened to the middle of a circular piece of leather, and the leather is moistened

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