reduced to powder in an agate mortar. An alkaline silicate is formed when this powder is heated with three times its weight of sodium or potassium carbonate. If this be added to an excess of diluted hydrochloric acid, the acid combines with the metal, and although the silica is thus ejected from its combination, it does not precipitate; but if heat be now applied, when the solution has reached a certain point of concentration, the silex is again rendered insoluble, and becomes a gelatinous mass. By still continuing the heat until much of the water is driven off, then the chloride of the metal may be washed out, and the pure silex again remains in its insoluble form. When rendered anhydrous, it is a white, light powder. All spring or well waters contain silica, either in a free state or in solution as an alkaline silicate, and by this means it enters into many vegetable and animal organisms. Silicon Chloride (SiC).-As the corresponding boror, compound, this is got hy heating cilica mixed with carbon in a current of chlorine. It is a liquid, and has the same remarkable action with water as the boron chloride, being decomposed into hydrochloric acid and silicic acid. Silicic Fluoride (SiF ̧) is the gas by the formation of which, with hydrofluoric acid, the etching of glass is effected. When There is also a sulphide of silicon, which is a solid. thrown into water it decomposes, sulphuretted hydrogen escapes, and soluble silicic acid is taken up by the water. METALS. GENERAL PROPERTIES, ETC., OF METALS. 66 It was stated in an early chapter that the distinction between a metalloid and a metal was not very decided. Perhaps the definition of a metal," which will be finally adopted, will be that it is "a body which, when in solution, is carried with an electric current which traverses that solution;" but on referring to the analysis of water by the voltameter, it will be noticed that hydrogen passes with the current; hence it is necessary, if this definition be adopted, to class that gas among the metals. There is no reason for doubting that if hydrogen could be condensed into a solid it would exhibit metallic lustre and the other properties common to all metals. The chief properties which characterise metals areMetallic Lustre.-This lustre, however, is not peculiar to the class, for it is shared by iodine and graphite. have been burnished. 30000 placing a wire of platinum in the axis of a bar of silver, and then in the usual way-that is, by drawing it through holes in a steel plate-procured a very fine wire, the centre of which was one much finer of platinum. By melting the silver off with nitric acid this was exposed, and found to be 50th of an inch in diameter. The foregoing qualities are greatly dependent on the texture of the metal, and the requisite texture, by various manipulations, may be procured. Thus cast-iron is brittle, so is hard steel; yet steel is capable of being made into watch-springs, and iron can be drawn into very fine wire. Specific Gravity is the relation which the weight a body bears to the weight of an equal volume of water. greatly in their specific gravities : The metals differ while platinum requires the intense heat of the oxyhydrogen Alloys. When metals enter into combination, "alloys" are formed. The union is now generally considered to be "chemical." The various alloys will be treated of under their metals. An "amalgam" is an alloy in which one of the metals is mercury. ascores. The Appearance of Metals in the Earth's Crust.-Gold, plain an uncombined form; but the last-named three are also found tinum, silver, mercury, and copper are found "native"—that is, Moreover, if most metals be procured in a state of fine sub-buted in nature. Lime is the oxide of calcium. The tint of the Next to silica, metallic oxides are most widely distri division they are lustreless. This is prominently the case with sandstones is due to the oxide of iron. gold, which, when painted on china, in a chemical solution, and tain many oxides, while the waters of the ocean are rendered The granite rocks conthen burnt, comes out of the furnace dull green, and the well- briny with their salts. But the metals which we require in known lustre of gold does not appear until the painted portions large quantities we seek from masses of their ores, which are found in mineral veins. It often appears that the older rocks have, in the convulsions of Nature, been rent into fissures. with molten basalt, trap, etc., thus forming dykes, they have When these openings have not been filled up from below been filled up with metallic ores, which seem to have been usually deposited from above. These constitute "lodes." The and, strange to say, if this direction be altered, the ore genelodes of a neighbourhood generally run in the same direction, rally alters its character, and frequently its chemical constitu Opacity.-Metals are usually considered opaque, but this is not absolutely true, for gold, if not more thanth of an inch thick, permits green light to pass through it, and other metals exhibit a similar imperfect opacity when reduced to very thin leaves. Hardness-Steel, which is a compound of carbon and iron, is the hardest of the metals, and the rest pass through every degree down to potassium, which may be moulded by the fingers as if it were putty. The hardness of a metal may be greatly increased if its temperature be raised to a certain point and then suddenly cooled by plunging it into water. Many of the hard metals are sufficiently elastic to be sonorous when struck. Brittleness and tenacity are closely connected. Many of the metals, such as bismuth, antimony, arsenic, are so brittle as to admit of being pounded in a mortar, and many, such as steel, iron, copper, can be made brittle by suddenly reducing their temperature. Others, again, exhibit this quality only at certain temperatures: for instance, zinc, which cannot be bent without danger of cracking, is readily worked at a temperature of about 115° Cent.; and on the other hand, brass-an alloy of zinc and copper becomes brittle as it approaches a red heat. Tenacity is the property which gives to metals their power of supporting a strain. Iron possesses it in the most eminent degree, hence the great value of this metal. Malleability and Ductility.-Possessing the one property, the metal may be beaten or rolled into very thin leaves; with the other, it admits of being drawn into wire. Gold, silver, copper, and platinum are very malleable. Gold may be beaten out into leaves so thin that 280,000 only make an inch. These metals, and iron, are the five which are notably ductile. Wollaston, by tion. This and other reasons have led to the belief that electricity has been the chief agent in depositing metallic veins. It sometimes happens, as in the case of the iron beds in our own country, that the ore is found in strata, occupying the position of a layer of rock. SKETCHING FROM NATURE.-I. MATERIALS-CHOICE OF SUBJECTS, ETC. In our Lessons in Drawing, to be found in the previous pages of the POPULAR EDUCATOR, we have endeavoured to place before our pupils the general principles which belong to and are applicable to the practice of drawing from the flat (that is, from copies), and also those principles which guide us in drawing from the object. We now undertake a more direct application of the instruction therein given, for the purpose of introducing our papils to that very interesting and delightful practice of drawing, usually termed "sketching from nature; " we mean by this, the taking up a few simple materials and seeking our subjects out of doors. The phrase "sketching from nature" is & very convenient one, and is generally understood, therefore whether it is easier or more difficult depends upon the inclination of the mind, the practical experience, or, speaking more exactly,! the kind of experience the pupil has been accustomed to. If the grammar of the art has been well learned, the pupil will find that a very considerable amount of the knowledge he has acquired whilst drawing from the flat will be of the greatest service when drawing from nature. We have frequently met with portrait painters who have had to make duplicates of their pictures, and who have said they would much rather paint them again from the sitter than copy them from the original picture: only those who have experienced it can fully understand how much more feeling and life can be imparted to the work when nature is the guide, than when they have to depend upon the limited expression of a copy. So with landscape: we have frequently been more pleased with the 66 Fig. 1. we will retain it, although we prefer the expression drawing from nature," as it implies greater care and attention to details than the term sketching in its usual sense. A loose habit of drawing may be called sketching, and if this were all that is understood by it, the practice would be a dangerous one for a beginner; but as we have already given sufficient cautions upon this point in the lessons upon Drawing, we will only repeat one piece of advice and pass on-"Learn to draw first; sketch afterwards." In the course of these lessons we shall find it necessary occasionally to refer back to the lessons in Drawing already given, as our object is to apply practically the principles which have been there stated. How many times has the question been asked, "Do you draw?" And what is the reply in the great majority of cases? "Yes, but only from copies; I have never attempted to do anything from nature, having always considered it so much more difficult." Now, there are those who maintain the reverse, namely, that drawing from nature is easier than copying pictures. Certainly the former is much more pleasant, and more satisfactory, as all must acknowledge; VOL. III. "original sketch," taken upon the spot, than with the finished picture painted from it in the studio at home. Although the "original sketch" was not so highly finished as the picture, yet it had the stamp of nature and freshness upon it, which could best be caught from the scene itself, and which it is diffi cult to impart at second hand. As the eye of the student becomes more and more accustomed to Nature, and keener to detect and appreciate her beauties, he will discover much of which a common observer has but an imperfect perception; to the latter, a landscape is the same to-day as it was yesterday, he can only see trees, buildings, and other objects abstractedly through one and the same medium; while the eye of the artist is continually discovering something fresh, perhaps principally caused by the successive changes of light, or from the positions of objects in relation to each other, and their contrasts in both colour and form. The tree before him in the morning may certainly be the same that he sees in the evening, but how very different is the effect, and what a multitude of details with all their beauties, which were imperceptible in the morning, are 59 brought out by the change of light. We have no doubt that many of our pupils, when they have conquered their early difficulties, will discover with pleasure and surprise that drawing from nature has a charm about it which cannot be realised by copying only. The necessary materials are simple :-A block, that is, a solid mass of paper composed of several layers bound together only at the edges, so that when a drawing is completed we have only to slip a penknife between it and the next paper, pass it round, remove the drawing, and underneath will be found another surface like the one already filled, ready for use. The kind of paper for pencil drawing ought not to be very rough, a slight grain will assist the pencil to mark freely, but on very rough paper it is impossible to give a very high finish to the work; rough papers are better adapted for colours. A few pencils, H, HB, and B, and a portable sketching stool, will be all that is requisite for our first essay. Being now prepared, let us suppose that we are on our way in search of a subject, and in the meantime we will make a few observations which especially apply to beginners. No one who has been accustomed to copy pictures only, can altogether comprehend what a very different thing it is to draw from nature until he has made the attempt, when he will discover there are several reasons for the difference. One is, that all the objects in the picture are reduced for him, probably to the exact size he wishes to make them; another is, the outline upon the paper has a more definite effect than the general form in nature, which admits of no actual boundary line, but presents only the mass discoverable from other objects by colour, and light and shade; another reason is, that objects in nature advance or recede from one another, whilst in a picture they are all arranged upon one plane or surface; and thus we are led to acknowledge the necessity of knowing something both of lineal and aërial perspective. It is true many depend upon the eye alone for the proportions of the retiring parts as they recede, and consequently are liable to make frequent and serious mistakes, which a little acquaintance with perspective would prevent; but we intend to take up this part of our subject again. We will now pass on to another consideration with reference to the choice of subject for the first attempt of a beginner. We well know the feelings with which most beginners go out for the first time to draw from nature; their enthusiasm would persuade them to attempt great things; nothing short of some extensive prospect, hill and dale, woods, rivers, buildingsin short, a whole country side. Upon this point we wish to caution our pupils. It is one of the first and greatest mistakes which young painters make when they begin to draw from nature; nearly all, without exception, sit down to take some extensive view, without a question as to its composition, and without any inquiry whether they will be able to go through with it. The principal reason they give for their choice is "the beauty of the scene." We knew a case some years ago of a young student in the Royal Academy, who copied in the painting school an elaborate landscape by an old master; succeeding beyond his expectations, he felt a strong desire to paint a picture from nature, having now, as he thought, acquired sufficient power to justify the attempt. Accordingly, he went to the top of Highgate Hill, and commenced a picture of the entire prospect looking northward; he worked hard for several days, but found he was alternately painting in and rubbing out; the constant changes of sunshine and shade, as they passed over the landscape, perfectly bewildered him, and the result was that he gave it up quite disheartened. He resolved, however, to show the little he had done to the late Mr. Constable (the painter of "The Corn-field" in the South Kensington Museum), and ask his advice. Mr. Constable looked first at the picture and then at the youth, and in a quiet way, though with unmistakable meaning, said, "My young friend, go and draw a gate-post, and when you have done that draw two posts, and go on till you can manage a dozen; afterwards add a cottage, then a tree, and proceed in this way until you have power to do something more elaborate before you think of painting such a subject as this. You have made precisely the same mistake that I made when I was your age; you have begun at the wrong end." The above excellent advice needs very little comment from It is exceedingly valuable, and forcibly suggests the folly of rushing headlong into a multitude of difficulties from which us. there is no escape, but at the cost of much discouragement. All must acknowledge that, whatever may be the extent of the subject they propose to draw, it is essentially composed of several particular objects, each of which requires a separate and careful study. Now the first question every one must ask himself should be, "Can I copy any one of these objects, independently of the rest?" If he cannot, let us assure him it would be useless to attempt the whole together. All who have reached any eminence in the art have found from experience the advantage of overcoming the difficulties connected with single objects first. Our pupils will clearly see from these remarks that the simpler the subject the better for a first trial, so that as their strength and confidence increase they will find themselves capable of enlarging their subjects, of entering more closely into their numerous details, and as they proceed a proportionate amount of increasing satisfaction will be gained, and the art itself will become more and more interesting. Fig. 1 will give some idea of the class of subject for a first attempt, and the manner of treating it, which need not be much beyond a carefully arranged and cleanly drawn outline; the shadows might be slightly marked in by a few parallel lines under the projecting parts, down the shadowed sides of the posts, to define and to bring forward the branch of a tree. In this simple arrangement of a few posts and weeds, there are no important retiring lines, consequently there will be no necessity for vanishing points, a subject for our consideration in another lesson. The distance of the station point, or the position of the draughtsman from an object of this class and extent, might be about a dozen or fourteen yards, because at that distance all contained within its outer limits will be considerably within an angle of 60°. See "Lessons in Drawing," Fig. 25, and the remarks upon it (Vol. I., page 72). Subjects of the class we have selected are very common: a stile, à bridge over a brook, and many more of the same kind, are to be found almost everywhere. We have just said that the drawing need not be more than a carefully arranged outline. If for some time the pupil will confine him. self to outline, and use no more shadow than is necessary to assist in making the form clear and intelligible, it will be an advantage, because it is doing one thing at a time, and he is not overpowering himself with difficulties; besides, shading bad outlines is a waste of time, as shading cannot improve the drawing, nor can it be successfully practised without the power of correct drawing, as it is only an additional help to represent the form marked out by the outline. There are other important considerations to be attended to. The pupil must remember, when he is seated, that the few moments before he puts his pencil on the paper are very important. First, he must decide how much of the subject he intends to draw; that being deter mined, he must fix upon the centre of the subject to be arranged in the centre of his paper, and as in most cases the eye will be considerably below the centre, there will then be sufficient room for the sky above, and the foreground below the object. Probably a single trial will induce him to make this a general rule until experience has taught him to arrange this matter for himself according to his position and the nature of the subject he is drawing. The next piece of advice we would give him before he begins, is to fix his whole attention upon what he is about to draw; he must examine not only of what it is com posed, but he must attentively observe how the several parts are arranged with regard to each other, and what are the rules and principles he has at command for his purpose. As he is about to draw it as it appears to him, without attempting any effect which does not strictly belong to it, he must take up one principle at a time; the first will be form-this refers, in the first instance, to the shape and character of the subject as a whole; then the position of the parts relative to each other; all important particulars must be carefully examined, his eye and his mind must become familiar with everything; this will strengthen his confidence, so that when he begins to draw, the acquaintance he has made with his subject will be of the greatest value. In practice, it is quite allowable to determine the relative heights of the parts with one another by placing the pencil horizontally before the eye, having its edge on a level with any particular point, and by looking along the remaining portion of the pencil when thus placed, the pupil will be able to see at once which other portions are on the same level, which are above, and which below; he must notice where lines if produced would cut other lines already drawn, and also where LESSONS IN ARITHMETIC.-XXXIX. FELLOWSHIP, AVERAGES, MIXTURES IN CERTAIN PROPORTIONS, ETC. 1. Ir several partners have invested different sums in an undertaking, it is manifest that the profits or losses must be divided among them in proportion to the capital each has invested in the business, if the capital of each has been in use for the same time. The method by which the share of each is determined in this case is called Simple Fellowship. It is manifestly the same as that given in Lessons in Arithmetic, XX., on Ratio and Proportion, Arts. 7, 8 (Vol. I., page 343), where a given number is divided in proportion to certain others. EXAMPLE.-A, B, and C put into a business £300, £500, and £800 respectively. At the end of a year they have gained £400. What is the share of each ? We have to divide £400 in the proportion of 300, 500, 800, or, what is the same thing, in the proportion 3, 5, 8. According to Lessons in Arithmetic-XX, Art. 7, since 3 + 5 + 8 = 16, we divide 400 into 16 equal parts, each of which is £25. The respective shares of A, B, and C will be £75, £125, and £200. 2. If the sum invested by each partner is not used for the same period of time, so that we have to take into account not only each sum, but the time during which it is employed, the case is called one of Double or Compound Fellowship. EXAMPLE.-A, B, and C contribute to a business as follows: A puts in £1,200 for three months, B £1,000 for six months, and C £800 for twelve months. How must they divide a profit of £800 ? A's €1,200 for 3 months is the same as 3 x £1,200, or £3,600 for 1 month. EQUILIBRIUM OF FLOATING BODIES-METACENTRE CAPILLARY ATTRACTION-HYDRAULIC MACHINES-WATER-WHEELS. WE have now examined at some length the effects produced on bodies by immersion in liquids, and have seen that one of the conditions of equilibrium is that the weight of the displaced fluid shall be equal to that of the body immersed. This, however, is not the only condition that must be complied with in order to ensure equilibrium. Suppose, for instance, that we have a solid of the shape of A B in the annexed figure, and that the end в has a piece of lead affixed to it, so as to render it heavier than the other. Let us now see what are the forces acting on this body, One force is its own weight, which acts through its centre of gravity G, and as the end в is heavier than the other, this point is nearer that end; the other force acting upon it is the buoyancy of the liquid which acts through the centre of gravity of the displaced liquid, that is, upwards through G'. Now these two forces are equal to one another, and act in opposite directions, but their lines of action do not pass through the same point; and hence, as we saw in our Lessons on Mechanics, they constitute "a couple," and produce a tendency in the body to twist round. In order, then, that there should be equilibrium, the points G and Gʻ must be in B's £1,000 for 6 months is the same as 6 x £1,000, or £6,000 for 1 the same vertical line. month. C's £800 for 12 months is the same as 12 x £800, or £9,600 for 1 month. Hence the shares must be in the proportion of 3,600, 6,000, 9,600; Le, of 3, 5, 8. Hence, procceding as in Simple Fellowship, the shares will be respectively £800, £800, £800; that is, £150, £250, £400, This example sufficiently explains the following Rule for Compound Fellowship. Multiply each capital by the number of units of time for which it is employed; the shares will be in the proportions of these products, and will be determined as in a case of Simple Fellowship. EXERCISE 60. 1. A traveller divided 80s. among 4 beggars in such a manner that as often as the first received 10s., the second received 9s., the third 8s., and the fourth 78. What did each receive? 2. A, B, and C engage in business, putting in respectively £3,500, £5,000, and £6,000. What would be the share of each out of a profit ct £1,000 ? 3. A, B, and C contribute respectively to a speculation £160, £240, and £480, and they gain £264. What will be the share of each ? 4 A, B, C, and D embark in a business, and put in respectively £2,000 for 6 mouths, £1,500 for 9 months, £1,000 for 12 months, and £750 for 15 months. If at the end of 15 months the profits are found to be £1,000, how must they be divided? 5. A and B form a partnership for a year. A contributes £5,000, to which at the end of 6 months he adds £1,000 more; B contributes 20,000, and at the end of 9 months withdraws £2,000 from the busi How must they divide a profit of £1,500 at the end of the year? 6 A, B, and C form a partnership; A contributing £1,000, B £2,000, and C £3,000. After 9 months C withdraws, and after 3 months more Dis admitted to the partnership, contributing £1,500. At the end of 18 months, the partnership being dissolved, the profits are found to be £900. How must they be equitably divided? Fig. 18. Now either G or G' may be the higher, and, according to this, A B is in a state of stable or unstable equilibrium. If the the body will remain at rest until some disturbing force acts centre of gravity of A B be above that of the displaced liquid, tendency to rotate will come into action, and the body will move on it; but as soon as it is moved at all from its position, the further and further from its original position. If, on the other hand, a' be above G, and the body be then deflected slightly from its position, the forces acting on it will draw it back. Hence it is said to be in a condition of stable equilibrium. In the case of floating bodies, or of vessels going to sea, it is clearly of the utmost importance to be sure that they are in this condition, as otherwise a little wind will cause them to incline, and they must then turn over. We will see, then, what are the conditions requisite to ensure safety. Fig. 19. downwards, and it would then remain at rest. Now let it be The body A B, in Fig. 18, would be found to turn till в was turned a little from its vertical position, as in Fig. 19. The dotted line represents its axis, in which both G and G' were situate, but in the new position G' will be at one side of this axis. Draw from this point a line vertically upwards to represent the buoyancy of the water, this line will cut the axis in some point, M. If this point be above G, the body is in a state of stable equilibrium; if it be below, the body is unstable. This point м is called the metacentre. Hence, if the metacentre be above the centre of gravity, the vessel will float in safety. Now from this we learn several important things. The first is, that in a vessel the centre of gravity should be as low down as possible. A captain accordingly arranges to stow the heaviest part of his cargo in the lowest part of the hold; and for the same reason, in a ship almost empty, or in a pleasure-boat, a large amount of heavy material, such as clay or pig-iron, is stowed away as ballast. If the lower part were left empty, or filled with light cargo, and heavy goods placed on the deck, the centre of gravity would be raised dangerously high, and the vessel, in all probability, would capsize. Forgetfulness of this fact is a fruitful source of danger to passengers in rowing or sailing boats. If a squall comes on, or any accident seems imminent, the passengers frequently spring to their feet, and by so doing greatly raise the centre of gravity and increase the danger; the wisest and safest plan is for all to sit down-or, better still, to lie down-at the bottom of the boat; the centre of gravity being thus lowered, the danger will be much diminished. Another thing that should be carefully seen to in sailing vessels is to have the cargo so stowed, that the centre of gravity is vertically over the keel, and also to prevent its shifting its position when the vessel lurches, as, if it does so, she cannot right herself so well. In paddle-wheel steamers, where it is important for the vessel to be upright, small carriages, filled with chain or other heavy material, are often placed upon the deck, so that when the wind inclines the vessel, these may be moved to the higher side, and thus bring it even again. Fig. 20. We now pass on to notice another property of liquids, known as capillary attraction. If we procure several glass tubes (Fig. 20) of small but different diameters, and dip them into water, we shall find an apparent exception to the rule that liquids maintain their level, for the water will rise in them to a height which varies with the size of the tube. This height increases inversely as the diameter. The name," capillary attraction," or capillarity, is derived from the Latin word capilla, which means a hair," and was so used because this effect was first observed in tubes almost as fine as a hair. We see a great many common things which afford illustrations of this fact. A lump of sugar consists of a large number of small crystals held together so as almost to touch, and they leave small tubes or passages between them. Hence, if we just dip a corner into a cup of tea, we see that the tea rises at once and wets the whole lump. A better illustration is to procure a tall lump of salt, and set it in a plate filled with some coloured fluid, as water and red ink; the line produced by the rise of the liquid is then very clearly seen. If a towel or piece of linen be placed in a vessel of water, a portion being allowed to hang over one side, it will in the same way draw up the water in its interstices and allow it to drip from the lower corner, thus emptying the vessel. A practical application is made of this principle in quarries where millstones are obtained. A block of stone is roughly trimmed to a cylindrical form. Grooves are then cut round it at distances regulated by the thickness of the stones. Into these grooves wedges of dry hard wood are firmly driven, and the damp of the air is so powerfully attracted into their pores that they swell and split off the stones from the block. On the same principle a candle burns. The heat of the flame melts the tallow or composition, and forms a cup filled with the melted portion; this rises in the wick by capillary attraction, and there it is converted into a gas, and consumed, while it gives light. In all these cases we have supposed the solid has been of such a nature as to be wetted by the liquid. If, however, this be not the case, the liquid in the tube will stand at a lower level than that without. This may be tried with a glass tube dipped into mercury, when the mercury within the tube will be seen to be at a lower level than that without. These effects are accounted for by the attraction or repulsion of the surface of the tube for the liquid, and may be seen well by immersing a sheet of glass in the liquid, or, better still, by taking two glass plates and moving them different distances apart. If we arrange them, as shown at Fig. 21, so that the edges at one side meet, while at the other they are a small distance apart, the liquid will rise between them and form a curve, Fig. 21. and thus we can ascertain how high it rises for each different distance between the plates. The elevation here is found to be just one-half of what it is in tubes having a diameter equal to this distance. These experiments you can try for yourselves, and by doing so will be far better able to understand them. Never be satis fied with reading an account of an experiment, or looking at an illustration of it, if you can try it. We have now noticed the main points in the first branch of our science, or Hydrostatics proper, and must pass on to the second branch, or that which treats of liquids in motion, and of the various modes of raising them, or deriving motion from them. HYDRAULICS. When a liquid is contained in any vessel it exerts a pressure against the sides, and this pressure we found to vary with the depth below the surface. If now we make an aperture in any portion of the side the liquid will rush out; and as the velocity with which it flows depends on the pressure, the lower down the aperture is situated the greater will be the velocity with which it flows. If we have a vessel of the shape shown in Fig. 22, with several jets inserted at different points along the side, which can be opened or closed at pleasure, we can ascertain how much flows from each, and from this the velocity with which it issues. It is, however, necessary to maintain the water at the same level during the experiment, and therefore a spout is made at A, and the vessel so placed that a stream of water from a tap runs in rather more rapidly than it issues from any of the jets. The surplus water will escape by the spout, B and thus maintain a uniform level and pressure. By a series of experiments conducted in this way Torricelli arrived at the conclusion that if the distance of any jet, E, below the surface is four times as great as that of any other, B, the velocity with which the water will issue from the first is twice as great as from the other; that is, that the velocity varies as the square root of the height of the water. Further experiments point out that this velocity is just that which, under the Fig. 22. laws of gravity, the liquid would acquire in falling from the surface to the opening. Thus, if the jet be one foot below the surface, the liquid will issue with a velocity of eight feet per second, that being the velocity a body acquires in falling through a space of one foot. If, then, the aperture have an area of 1 square inch, 96 cubic inches ought to flow out in one second, but on trying the experiment we find that only about 60 cubic inches actually flow, or about 62 per cent. of the calculated amount. This discrepancy seems at first sight to show the inaccuracy of the law, but on further examination it only confirms it. If we make an opening in the bottom of a vessel (Fig. 23), and carefully observe the water as it issues from it, we shall notice that the stream is not the same size throughout, but narrows consi. derably just beyond the orifice, so that the smallest area is a little way below the opening. Thus, if A в be the aperture, the part of the stream with the smallest sectional area will be at a b. The particles of water, in flowing along towards the opening, acquire an onward motion, which they retain as they flow out, and this narrows the stream. Now the section at ab is found to be just of that at AB, but the actual efflux we found was actually this fraction of the theoretical. If, then, we take as the area through which the water flows, not that of the aperture, but that of the section at a b, the actual flow will just correspond with the calculated amount. The diminished stream at a b is called the vena contracta, or contracted vein, and in all their calculations allowance is made by hydraulic engineers for the existence of this. We have thus far supposed the water to flow from a hole made in the side of the vessel; the actual amount that escapes is, however, very much varied by inserting a jet or pipe from which the water may issue. If a straight pipe, whose length is about three times its diameter, be inserted in the opening, the flow from it will be increased to about 82 per cent. of the theoretical amount; if this pipe be slightly tapering outwards, the issue will be still greater, while, if it taper inwards, the external |