(2.) What is the product of aa X aaa X aaaa by aaa X aaaa? Ans. aaaaaaaaaaaaaaaa, or ale.

79. It is also plain, from Art. 73, that the numeral co-efficients of several factors should be brought together and made into one factor by multiplication. Thus to multiply 2a x 3b by 4a X 5b, gives the product of 2a x 36 x 4a x 5b, or 120aabb; For the co-efficients are factors [Art. 24], and it is immaterial in what order these are arranged. Therefore 2a X 3b X 4a X 5b =2 X3 X4 X5 XaXaxbxb= 120aabb.

instead of being allowed to run in its natural bed, is stopped by a dam or embankment built across it. The water accumulates behind this till it rises to

2y X 4? Ans. 480xxyyz.

(2.) What is the product of 3a X 4bh by 5m X 6y? 360abhmy.

(3.) What is the product of 46 x 6d by 2x + 1? 48bde+24bd.

80. The product of two or more powers of the same quantity is expressed by writing that quantity with an index equal to the SUM of the indices of the proposed powers. Thus the product of a2 and as is as; and the continual product of x3, x, and as is a13. So likewise the product of am and an is am+", and that of x and is an+1; and, on the same principle, the product of am-n and 2 is . The reason of this is evident from Art. 79. Thus a and a3 are the same as aa and aaa; the product of which is aaaaa or as; the index 5 being the sum of the indices 2 and 3, the numbers which show how often a is used as a factor in the given powers.

EXAMPLES.-(1.) What is the product of a2 and as? Ans. a". (2.) Find the continued product of a2, ab, and a4b2? Ans. a7b3. (3.) Find the continued product of x, y, xy3, and xy. Ans. 10.

LESSONS IN HYDROSTATICS.-VI.
BREAST WHEELS-OVERSHOT WHEELS-TURBINES-BARKER'S

MILL-HYDRAULIC RAM-MACHINES FOR RAISING WATER
-PERSIAN WHEEL-ARCHIMEDIAN SCREW.

We examined in our last lesson the principle, and saw some of the advantages and disadvantages, of the undershot waterwheel. There is one other advantage, however, which should be just noticed, and that is that when made in its simplest form, with the floats radiating directly from it, it can be used in a tidal stream.

In any other kind of wheel, the water must flow in one uniform direction. The level should also be nearly even, as the height at which the water strikes the wheel makes a great difference in its working. With this kind, however, we have merely to fix the wheel between moored barges, so that it may rise and fall with the tide, and we shall have it almost constantly at work, part of the time turning in one direction, and part in the other. Only a small portion of the force of the stream will, it is true, be utilised; but, as there is usually in tidal rivers very much more power than is needed, this is of

Fig. 25.

it, in which the wheel is placed, with its floats nearly touching the sides, so that little water can escape. Much therefore depends upon the accuracy of the workmanship; if the water leaks by between the floats and the brickwork, there is a corresponding loss of power; while, on the other hand, if the floats scrape against the sides, there is a loss by friction. In this wheel the floats are frequently curved or bent in the middle, so as to hold the water better. It is important to take care that the water, after it has left the wheel, flows away without obstructing its motion, as a great loss of power sometimes results from this not being done. A step is frequently put in the course, as shown in the illustration, so that, as soon as the water has reached the lowest point of the wheel, it falls out of contact with it. Where this cannot well be done, part of the water is allowed to run by a channel at the side of the wheel, and by its momentum produces a current which aids in carrying away the tail-water.

In the overshot wheel, which is represented in Fig. 26, the water is usually diverted from the stream and conducted by a series of troughs to the top of the wheel, where it pours into the buckets. The end of the trough is sometimes open, so that the water flows on with the impulse it has acquired; the better plan, however, is for it to be closed, and an opening made underneath, as shown in the figure, through which the water may flow.

In this way it falls on the wheel without any momentum, and all strain is avoided. With this wheel nearly all the power of the water may be employed, as it may be made of such a width that none of the stream runs to waste. It is found best to let the wheel turn very slowly, as thus there is very little momentum left in the water when it leaves the wheel, and all loss from the water being thrown off by centrifugal force is avoided. Cogs are, therefore, usually placed round one edge of the wheel; these work in a pinion, and the motion is imparted to the machinery from this pinion instead of from the axis of the wheel. The shape of the buckets is, perhaps, of more importance in this than in the other descriptions of wheel, for, since the weight of the water is the moving power, it is important to retain as much of it as possible in the buckets until they arrive at the lowest point. Sometimes they are made of the shape shown in the figure, and masonry erected to confine the water, as in the case of the breast wheel, but more frequently

Fig. 26.

the buckets are made to curve upwards, and thus retain the water.

Another matter has, however, to be considered in deciding on the best shape of bucket, and that is to allow the air to escape. As the stream pours in it has to displace the air, and with some shapes of bucket this opposes and scatters the water in a much greater degree than would at first be expected. Openings are sometimes made in the cylinder for this purpose. In the best

constructed wheels of this class, about 80 per cent. of the power may be utilised.

Having thus noticed the wheels with horizontal axes, we must turn to those whose axes are vertical. Some of these are now coming into much more frequent use, being in many respects superior to those we have been considering, as they utilise a larger portion of the force, occupy less room, and work under water.

The most useful and generally used of these is the turbine. A cylinder is constructed with an opening round its lower portion, through which the water is allowed to flow from the higher to the lower level. Surrounding this opening, the width of which can be altered at pleasure, is a horizontal wheel with curved floats. This wheel is under water, and is supported by a plate on which it rests, its axle passing up through the cylinder. The water as it issues from the cylinder strikes against the floats, and thus turns the wheel with very great speed. Perhaps the best idea we can form of it, without actually seeing one at work, is to imagine a water wheel with curved floats to be turned on its end, and the internal cylinder removed, leaving the floats supported on the lower end, and held in their place above by a ring, to which they are fastened. The water is then allowed to flow into the interior of the wheel, and in its rush to escape between the floats it strikes upon their curved surfaces, and thus sets the wheel in motion.

Fig. 27.

In order that the water, instead of flowing directly from the centre, may strike the floats at a more advantageous angle, the cylinder is divided into compartments by means of curved partitions after the plan shown in Fig. 27, which represents an horizontal section of the cylinder and wheel, the outer ring being the wheel. By this it will be seen that the water issues in a direction almost perpendicular to the surface of the floats, and thus produces the greatest effect. As the water enters at the inner side of the wheel and is given off from the exterior, there is little loss from currents in the water; the pressure, too, on the axle being equal in every direction, there is not a large amount of friction as there is in water-wheels where the pressure acts only on one side.

The mode of construction just explained gives the best idea of the principles on which the turbine acts, but many important alterations and modifications have been introduced which render the machine more useful. In some, the water, instead of issuing from the sides of the cylinder, flows from an opening or a series of openings in the bottom, on to a wheel whose floats are curved vertically, the construction being then similar to that shown in Fig. 28. A still further improvement on this is effected by allowing the water to enter from below instead of from above, as in this way, instead of increasing the pressure on the bearings, it in a great degree removes it. As, however, all different makers of turbines have special plans, differing more or less, and we have seen the general principle on which all act, it is unnecessary to describe minor details.

Fig. 28.

Another wheel which is much used in France is called the spoon wheel; several arms having somewhat of a spoon shape radiate from a centre to which they are fixed. These are so inclined that when the water issues from the trough along which it flows, it strikes them almost at right angles to their surface, and hence imparts a rapid rotation to the wheel.

There is another apparatus by which motion can be derived from falling water, which is frequently exhibited as a lecture

table experiment, but seldom, if ever, used in practice. It is known by the name of Barker's Mill, or the Wheel of Recoil, and is represented in Fig. 29. M is a glass vessel, mounted so as to turn upon an axle, and capable of being filled by means of a stopcock at the top. If used practically it would, of course, be so shaped that the water could enter here as rapidly as it issued below. At the lower part two arms, cc, are inserted. These are turned round at the end as shown at A, so that the water can issue from them, and in so doing, by its recoil, it causes the vessel м to rotate on its axis. Every part of the inner surface of the tube sustains a pressure produced by the water in the vessel, but the pressure on opposite sides of the tube is equal, and therefore no motion ensues from it, the opposite pressures neutralising each other. As, however, a portion of the surface is removed at A, the pressure on the part opposite to it is not balanced, and therefore causes motion. This mill is, however, at present only an ingenious scientific toy, nor does there seem much probability of its ever coming into use.

Fig. 29.

A very ingenious and useful piece of apparatus, invented by a celebrated Frenchman named Montgolfier, must be noticed here, as not only is it very useful, but it involves several of the principles we have already considered. It is frequently required to raise a quantity of water to some elevation, and this machine, which is called the Hydraulic Ram (Fig. 30), accomplishes this by the momentum acquired by the fall of a current of water. As a considerable fall of water is desirable, and it is likewise important that it should remain nearly constant, a dam is usually built across the stream, so as to form a reservoir, which overflows when full, and thus maintains a uniform level. From this reservoir pipes are brought along the bed of the stream to join on to D. The other end of this pipe is closed. A valve opening downwards is, however, inserted at c, near the end. The spindle of this passes through a guide, so as to keep it vertical, and it falls of its own weight when the pipe is empty; the pressure of the water, however, closes it. Weights are now placed on the upper end of the spindle, so that when the pipes are full of water pressing upwards, the valve only just opens from its weight. Another opening is made in the pipe at F. communicating with a large reservoir, A, from the upper part of which issues the pipe, B, by which the water is to be raised. The upper part of this reservoir is filled with air, and a small valve, not shown in the figure, is so placed as to allow a small additional quantity to enter from time to time, and replace that carried away by the water, which under pressure absorbs a small amount of it.

The opening between the pipe A B and the reservoir is closed by the valve F, which rises by the inward pressure of the water and is closed by its own weight.

We will now suppose the machine to be set in action. The weight on the valve at c being more than sufficient to overcome

Fig. 30.

the pressure of the water, the valve opens, and the water escap and runs to waste. That in the pipe, however, acquires imm diately a small amount of momentum, which enables it to rai the valve and thus close the opening. The momentum th acquired by the water cannot be instantaneously destroyed, a

would burst open the end of the pipe were it not for the valve at F. This provides an escape, and the water opens it, and causes a certain amount to enter the reservoir, compressing the air contained in it, and thereby forcing a fresh amount of water up the tube B.

The compressed air, however, acts as a spring, and thus the momentum of the column of water is soon destroyed, F then closes of its own weight, and the water in the tube being now at rest, c again opens and allows the water to escape as at first. When the weight at c is carefully adjusted this opening and closing succeed one another rapidly, producing a series of stoppages, by each of which a small quantity of water is raised in the pipe B.

A larger amount, however, escapes at c than ascends in B, and the amount raised diminishes, of course, with the height to which it is raised; still it is calculated that about 60 per cent. of the power of the water may be utilised by the arrangement, which certainly by its ingenuity reflect: great credit on the inventor.

Occasionally, in mines, a stream of water is caused to move an engine, constructed on exactly the same principle as the steamengine, the motive power being the pressure of the water instead of the pressure of steam. By an arrangement of valves the water

which have not yet been explained; but it is best to consider all together, as in this way we can better understand their differences in construction.

First, then, we notice those which act mechanically. The plan of raising water by means of a single bucket would naturally suggest the idea of fixing several one below the other, and thus an endless chain of buckets passing over a wheel at the top was constructed.

is made to press alternately on the upper and lower sides of the piston, and the motion thus produced is by means of a crank and fy-wheel communicated to the machinery.

We have thus noticed all the most important machines designed to derive motion from a fall of water, and now pass on to the second class, or those which are intended to raise water to any required elevation.

The buckets are brought up full, and when they reach the wheel strike against a support, and being turned over discharge their contents into a channel prepared to receive them. The wheel in this case may be turned by the foot, as is frequently done, or the power of animals may be employed.

The next modification of this arrangement is what is known as the Persian Wheel, which is represented in Fig. 31. Floats are fixed to one side of an undershot or tidal wheel, and in the other side of the rim are fixed a number of pegs, from which buckets are suspended. As the wheel is turned by the force of the current, these successively dip into the water, and are brought up nearly full. The weight of their contents keeps them in a vertical position till they reach the top, where they strike against trough, and thus are emptied into

a

it. The water is conveyed from this by a channel not shown in the figure. By this plan the water cannot well be lifted to any great height, as the diameter of the wheel must be greater than the height. This machine can be used in a tidal river, as it will work in either direction.

A further supply of water is, in this wheel, raised to the level of the axis on a totally different principle. The spokes of the wheel, instead of being made straight, as is the case in ordinary wheels, are hollow and curve considerably. Openings will be perceived on the rim, by which the water enters when they are

Water is one of the prime necessaries of life, and as its tendency is always to sink to the lowest level, various plans of raising it have been tried from the very earliest ages. The mest primitive is by means of a bucket fastened to a rope; after A time, it was found more convenient, when the height to which the water had to be raised was not great, to fix this rope to one end of a lever supported near the middle on crossed poles, and pull by means of a rope fastened to the other and shorter end. A further improvement on this, which is a the present day much used on the banks of the Nile, consisted in iring a weight at the other end of the lever, so as nearly to balance the bucket of water; a man then alternately raises and lowers it by pulling the rope. Much of the land in Egypt is irrigated by this contrivance, which is known as the Shadoof.

The common windlass is used instead of this where the water has to be raised from a great depth; as, however, there are a large number of machines in use, it will be best to make a simple division of them, and perhaps the simplest we can make is the following:

1. Those which act mechanically;

2. Those which act by the pressure of the air; 3. Those which act by centrifugal force.

The second of these divisions contains the common pump and similar machines, which, strictly, ought not to be explained till we come to treat of pneumatics, as they involve principles

immersed, and from the shape of the spokes it cannot flow out again, since the openings are higher than the bends. The water, therefore, travels along them towards the axis, and there is discharged into a trough prepared for it.

Another very ingenious and elegant machine, acting on the same principle as the spokes in the Persian wheel, was invented by the celebrated philosopher Archimedes, and is called after him the Archimedian Screw (Fig. 32). It consists of an inclined axis, which may be turned by a winch. One end of this is in the stream or reservoir from which the water has to be raised, and the other over the reservoir into which it is required to flow.

A tube or pipe is twisted spirally round this axle, the angle at which it is twisted being so arranged that as it is turned by the handle the water constantly flows towards the upper end. A glance at the illustration will show that the portions of the spiral on the side shown all incline to the right, so that the water in them flows in that direction. When used in practice, instead of a tube being twisted in this way, a spiral flange, like the thread of a screw, but projecting to a much greater distance, is fixed on the axis and made to turn inside a straight tube which it just fits.

In this way there is much less friction of the water, and a

Having thus noticed the wheels with horizontal axes, we must turn to those whose axes are vertical. Some of these are now coming into much more frequent use, being in many respects superior to those we have been considering, as they utilise a larger portion of the force, occupy less room, and work under water.

The most useful and generally used of these is the turbine. A cylinder is constructed with an opening round its lower portion, through which the water is allowed to flow from the higher to the lower level. Surrounding this opening, the width of which can be altered at pleasure, is a horizontal wheel with curved floats. This wheel is under water, and is supported by a plate on which it rests, its axle passing up through the cylinder. The water as it issues from the cylinder strikes against the floats, and thus turns the wheel with very great speed. Perhaps the best idea we can form of it, without actually seeing one at work, is to imagine a water wheel with curved floats to be turned on its end, and the internal cylinder removed, leaving the floats supported on the lower end, and held in their place above by a ring, to which they are fastened. The water is then allowed to flow into the interior of the wheel, and in its rush to escape between the floats it strikes upon their curved surfaces, and thus sets the wheel in motion.

Fig. 27.

In order that the water, instead of flowing directly from the centre, may strike the floats at a more advantageous angle, the cylinder is divided into compartments by means of curved partitions after the plan shown in Fig. 27, which represents an horizontal section of the cylinder and wheel, the outer ring being the wheel. By this it will be seen that the water issues in a direction almost perpendicular to the surface of the floats, and thus produces the greatest effect. As the water enters at the inner side of the wheel and is given off from the exterior, there is little loss from currents in the water; the pressure, too, on the axle being equal in every direction, there is not a large amount of friction as there is in water-wheels where the pressure acts only on one side.

The mode of construction just explained gives the best idea of the principles on which the turbine acts, but many important alterations and modifications have been introduced which render the machine more useful. In some, the water, instead of issuing from the sides of the cylinder, flows from an opening or a series of openings in the bottom, on to a wheel whose floats are curved vertically, the construction being then similar to that shown in Fig. 28. A still further improvement on this is effected by allowing the water to enter from below instead of from above, as in this way, instead of increasing the pressure on the bearings, it in a great degree removes it. however, all different makers of turbines have special plans, differing more or less, and we have seen the general principle on which all act, it is unnecessary to describe minor details.

Fig. 28.

As,

Another wheel which is much used in France is called the spoon wheel; several arms having somewhat of a spoon shape radiate from a centre to which they are fixed. These are so inclined that when the water issues from the trough along which it flows, it strikes them almost at right angles to their surface, and hence imparts a rapid rotation to the wheel.

There is another apparatus by which motion can be derived from falling water, which is frequently exhibited as a lecture

known by the name of Barker's Mill, or the Wheel of Recoil, and is represented in Fig. 29. M is a glass vessel, mounted so as to turn upon an axle, and capable of being filled by means of a stopcock at the top. If used practically it would, of course, be so shaped that the water could enter here as rapidly as it issued below. At the lower part two arms, cc, are inserted. These are turned round at the end as shown at A, so that the water can issue from them, and in so doing, by its recoil, it causes the vessel м to rotate on its axis. Every part of the inner surface of the tube sustains a pressure produced by the water in the vessel, but the pressure on opposite sides of the tube is equal, and therefore no motion ensues from it, the opposite pressures neutralising each other. As, however, a portion of the surface is removed at A, the pressure on the part opposite to it is not balanced, and therefore causes motion. This mill is, however, at present only an ingenious scientific toy, nor does there seem much probability of its ever coming into use.

Fig. 29.

A very ingenious and useful piece of apparatus, invented by a celebrated Frenchman named Montgolfier, must be noticed here, as not only is it very useful, but it involves several of the principles we have already considered. It is frequently required to raise a quantity of water to some elevation, and this machine, which is called the Hydraulic Ram (Fig. 30), accomplishes this by the momentum acquired by the fall of a current of water. As a considerable fall of water is desirable, and it is likewise important that it should remain nearly constant, a dam is usually built across the stream, so as to form a reservoir, which overflows when full, and thus maintains a uniform level. From this reservoir pipes are brought along the bed of the stream to join on to D. The other end of this pipe is closed. A valve opening downwards is, however, inserted at c, near the end. The spindle of this passes through a guide, so as to keep it vertical, and it falls of its own weight when the pipe is empty; the pressure of the water, however, closes it. Weights are now placed on the upper end of the spindle, so that when the pipes are full of water pressing upwards, the valve only just opens from its weight. Another opening is made in the pipe at F communicating with a large reservoir, A, from the upper part of which issues the pipe, B, by which the water is to be raised. The upper part of this reservoir is filled with air, and a small valve, not shown in the figure, is so placed as to allow a small additional quantity to enter from time to time, and replace that carried away by the water, which under pressure absorbs a small amount of it.

The opening between the pipe A B and the reservoir is closed by the valve F, which rises by the inward pressure of the water and is closed by its own weight.

We will now suppose the machine to be set in action. The weight on the valve at c being more than sufficient to overcome

Fig. 30.

H

the pressure of the water, the valve opens, and the water escapes and runs to waste. That in the pipe, however, acquires immediately a small amount of momentum, which enables it to raise the valve and thus close the opening. The momentum thus acquired by the water cannot be instantaneously destroyed, and