matter what be the length or shape of the plane, or whether it be straight, or curved, or spiral, for in all cases, if it only be smooth and of the same vertical height, you will get the same amount of energy by causing the kilogramme to fall from the top to the bottom.

41. But while the energy remains the same, the time of descent will vary according to the length and shape of the plane, for evidently the kilogramme will take a longer time to descend a very sloping plane than a very steep one. In fact, the sloping plane will take longer to generate the requisite velocity than the steep one, but both will have produced the same result as regards energy, when once the kilogramme has arrived at the bottom.

W

Fig. 1.

Functions of a Machine.

42. Our readers are now beginning to perceive that energy cannot be created, and that by no means can we coax or cozen Dame Nature into giving us back more than we are entitled to get. To impress this fundamental principle still more strongly upon our minds, let us consider in detail one or two mechanical contrivances, and see what they amount to as regards energy.

Let us begin with the second system of pulleys. Here we have a power P attached to the one end of a thread, which passes

over all the pulleys, and is ultimately attached, by its other extremity, to a hook in the upper or fixed block. The weight w is, on the other hand, attached to the lower or moveable block, and rises with it. Let us suppose that the pulleys are without weight and the cords without friction, and that w is supported by six cords, as in the figure. Now, when there is equilibrium in this machine, it is well known that w will be equal to six times P; that is to say, a power of one kilogramme will, in such a machine, balance or support a weight of six kilogrammes. If P be increased a single grain more, it will overbalance w, and P will descend, while w will begin to rise. In such a case, after P has descended, say six metres, its weight being, say, one kilogramme, it has lost a quantity of energy of position equal to six units, since it is at a lower level by six metres than it was before. We have, in fact, expended upon our machine six units of energy. Now, what return have we received for this expenditure? Our return is clearly the rise of w, and mechanicians will tell us that in this case w will have risen one metre.

But the weight of w is six kilogrammes, and this having been raised one metre represents an energy of position equal to six. We have thus spent upon our machine, in the fall of P, an amount of energy equal to six units, and obtained in the rise of w an equivalent amount equal to six units also. We have, in truth, neither gained nor lost energy, but simply changed it into a form more convenient for our use.

43. To impress this truth still more strongly, let us take quite a different machine, such as the hydrostatic

Fig. 2.

press. Its mode of action will be perceived from Fig. 2. Here we have two cylinders, a wide and a narrow one, which are connected together at the bottom by means of a strong tube. Each of these cylinders is provided with

a water-tight piston, the space beneath being filled with water. It is therefore manifest, since the two cylinders are connected together, and since water is incompressible, that when we push down the one piston the other will be pushed up. Let us suppose that the area of the small piston is one square centimetre,* and that of the large piston one hundred square centimetres, and let us apply a weight of ten kilogrammes to the smaller piston. Now, it is known, from the laws of hydrostatics, that every square centimetre of the larger piston will be pressed upwards with the force of ten kilogrammes, so that the piston will altogether mount with the force of 1000 kilogrammesthat is to say, it will raise a weight of this amount as it ascends.

Here, then, we have a machine in virtue of which a pressure of ten kilogrammes on the small piston enables the large piston to rise with the force of 1000 kilo

* That is to say, a square the side of which is one centimetre, or the hundredth part of a metre.

grammes. But it is very easy to see that, while the small piston falls one metre, the large one will only rise one centimetre. For the quantity of water under the pistons being always the same, if this be pushed down one metre in the narrow cylinder, it will only rise one centimetre in the wide one.

Let us now consider what we gain by this machine. The power of ten kilogrammes applied to the smaller piston is made to fall through one metre, and this represents the amount of energy which we have expended upon our machine, while, as a return, we obtain 1000 kilogrammes raised through one single centimetre. Here, then, as in the case of the pulleys, the return of energy is precisely the same as the expenditure, and, provided we ignore friction, we neither gain nor lose anything by the machine. All that we do is to transmute the energy into a more convenient form-what we gain in power we lose in space; but we are willing to sacrifice space or quickness of motion in order to obtain the tremendous pressure or force which we get by means of the hydrostatic press.

Principle of Virtual Velocities.

44. These illustrations will have prepared our readers to perceive the true function of a machine. This was first clearly defined by Galileo, who saw that in any machine, no matter of what kind, if we raise a large weight by means of a small one, it will be found that the small weight, multiplied into the space through which it

D

is lowered, will exactly equal the large weight, multiplied into that through which it is raised.

This principle, known as that of virtual velocities, enables us to perceive at once our true position. We see that the world of mechanism is not a manufactory, in which energy is created, but rather a mart, into which we may bring energy of one kind and change or barter it for an equivalent of another kind, that suits us better— but if we come with nothing in our hand, with nothing we shall most assuredly return. A machine, in truth,. does not create, but only transmutes, and this principle will enable us to tell, without further knowledge of mechanics, what are the conditions of equilibrium of any arrangement.

For instance, let it be required to find those of a lever, of which the one arm is three times as long as the other. Here it is evident that if we overbalance the lever by a single grain, so as to cause the long arm with its power to fall down while the short one with its weight rises up, then the long arm will fall three inches for every inch through which the short arm rises; and hence, to make up for this, a single kilogramme on the long arm will balance three kilogrammes on the short one, or the power will be to the weight as one is to three.

Fig. 3.

LENGTH

HEIGHT