case has been examined with much attention. At the hole the particles of liquid do not move vertically downwards so as to form a cylindrical column, but the lines of direction of the motion are inclined towards each other as if they were about to meet at a point. Thus the stream of issuing liquid is narrowest at a short distance from the hole, and this part of the stream is called the vena contracta or contracted vein. The area of a section of the vena contracta is equal to about five-eighths of the area of the hole. If in calculating the amount of liquid which would pass in a given time through a hole in the base of a vessel we take the area of the vena contracta instead of the area of the real hole, the result is found to agree reasonably well with observation.

XL. RESISTANCE OF LIQUIDS.

470. The resistance which a solid body experiences in moving through a liquid is a matter of great importance in practice; but the subject is not one which admits of elementary exposition, and we shall confine ourselves to a few simple remarks.

471. Suppose that a flat board is urged through a liquid which is itself at rest; suppose the board to move with uniform velocity in a direction at right angles to its plane. Then it is found by theory that the resistance which the board experiences from the liquid is at right angles to the board, and is equal to the weight of a column of the liquid which has the board for base, and for height the space through which a body must fall freely in order to acquire the velocity. The height by Art. 127 is equal to the square of the velocity divided by 64. But this theoretical result is not very exactly confirmed by experiment.

472. If the preceding result be accepted as correct, we see that we must apply to the board a force equal to the weight of the column there mentioned in order to keep it moving uniformly. For then the force which we apply just balances the resistance, and the board continues to move with uniform velocity according to the First Law of Motion. One fact involved in this result deserves to be

explicitly noticed: suppose a force to be applied just sufficient to keep the board moving at a certain uniform rate, then if we wish to have the velocity doubled we must exert four times as much force, if we wish to have the velocity tripled we must exert nine times as much force, and so on. For according to the statement of Art. 471, if the velocity is doubled the resistance becomes four times as great, and so on. Moreover some reason may be given in explanation. If the velocity of the moving board is doubled then the board strikes against twice as many particles of liquid as before in a given time, and also strikes each particle with twice the velocity it did before. Thus the board may naturally produce four times the movement in the liquid which it did before, and so may itself experience four times the resistance which it did before.

473. Next suppose that the board is urged through the liquid in a direction which is not at right angles to its plane. Suppose for instance that the board faces North East, but that it is urged in the direction from South to North. In this case the resistance of the liquid is exerted as before at right angles to the board, and its amount is found by resolving the velocity of the board into two components, namely, one at right angles to the board, and the other along the board; the former component is alone regarded, and the resistance at right angles to the board is the same as would be experienced by the board if it were moving in this direction with this component velocity. When we have thus obtained the resistance in the direction at right angles to the board, we may often have to consider only that part of it which acts in the direction of the motion of the board. The whole process is somewhat beyond the range of this book; but the important principle still holds that if the velocity is doubled the resistance becomes four times as great, and so on.

474. We can thus understand the difficulty which occurs in attempting to give a very great velocity to bodies moving in the water, as ships or steam-boats. As long as we keep to the same steam-boat then in order to double the velocity, supposed uniform, we must apply four times the force, and so on. Much may be done by trial in devising the most favourable shape for the steam-boat in order to diminish

the resistance, but still if we attempt to obtain a very great velocity the resistance becomes too great to be overcome with due economy in the use of force.

XLI. GASEOUS BODIES.

475. We have hitherto been explaining the properties of liquids, that is of fluid bodies which, although not absolutely incompressible, yet retain their dimensions practically unchanged under all forces to which they are usually exposed. In liquids the two opposing principles, cohesión and repulsion, may be said to be nearly balanced. In air and other gaseous bodies the repulsive principle prevails, so that cohesion seems scarcely to exist. The constituent particles of the body fly asunder if left unconfined, and require to be constrained completely in some manner if we wish to keep them before us for examination. They can be compressed by suitable force to almost any extent, and when the force is withdrawn they return to their original dimensions. They are frequently called elastic fluids.

476. The distinction between solid, liquid, and gaseous is not so much a distinction between bodies as a distinction between the different states which the same substance may assume, We know for instance that the same substance may be solid in the state of ice, liquid in the state of water, and gaseous in the state of steam, Chemists have

strong reason for believing that all bodies can be made to pass into these three states, and that the state assumed depends principally on the quantity of heat which is present. Gaseous bodies are sometimes divided into two classes; to one of these the term gases is more peculiarly appropriated, and to the other the term vapours. A vapour is a gaseous body which passes easily by a reduction of temperature, or an increase of pressure, into the liquid state; thus steam is a vapour because by a slight cooling it is reduced to water. A gas, strictly so called, retains that form under all ordinary conditions of temperature and pressure; thus carbonic acid is a gas because it is only by special means that it can be reduced to a liquid: and common air is a still more eminent example, because no means have yet been found for bringing it to another state.

477. The passage from the liquid to the gaseous state is usually accompanied by a large increase of volume. Thus a cubic inch of water is converted by boiling into about 1700 cubic inches of steam; so that the cubic inch of water becomes nearly a cubic foot of steam. If we suppose that the substance consists of a large number of particles, placed at nearly equal distances, then we may imagine that in passing from the state of water to that of steam the average distance between two adjacent particles becomes about twelve times as great as at first.

478. The changes of state take place at different temperatures for different bodies. Thus to cause water to take the solid state of ice the temperature must be reduced to 32 degrees of Fahrenheit's thermometer; while if the temperature is raised to 212 degrees the water becomes steam. Mercury freezes at about 40 degrees below the zero of Fahrenheit's thermometer, and boils at about 650 degrees. Thus we see that one substance, as mercury, may remain in the liquid state at a temperature so low that another substance, as water, becomes solid; and at a temperature so high that another becomes gaseous.

479. There is a curious fact connected with the passage of bodies from the solid to the liquid state, and from the liquid to the gaseous; it is expressed by the statement that in these changes heat becomes latent. Suppose that a pound of water at 32 degrees of heat as measured by Fahrenheit's thermometer is mixed with a pound of water at 174 degrees; it is found that the temperature of the mixture is 103 degrees which is half the sum of 32 and 174 degrees: the hotter water has lost, and the colder has gained 71 degrees of temperature. But now suppose that a pound of ice at 32 degrees is put with a pound of water at 174 degrees; after a time the ice will all be melted, and the temperature of the mixture will be only 32 degrees. The water has lost 142 degrees of temperature, and the ice has been melted without any apparent increase of temperature the heat thus lost by the water is said to be latent in the melted ice. Thus we see that in the process of converting a solid into a liquid a large quantity of heat is required which is in some manner absorbed by the liquid and does not become apparent by a rise of temperature.

In like manner when water is converted into steam a large quantity of heat becomes latent. The amount is greater than in the case of liquefied ice, being now about 900 degrees instead of 142. These numerical values have been differently assigned by various experiments; but extreme accuracy is unnecessary for our purpose.

480. The term latent heat has been long in use and perhaps does not often lead to any confusion or error; but there is always a danger that such descriptive terms should be made to suggest more than they are actually intended to convey. It might be objected in the present case that heat cannot be properly said to be hidden because its influence is manifested in the remarkable change of state, namely, from the solid to the liquid state, or from the liquid to the gaseous.

481. Common air is the most obvious and the most important of the gaseous bodies, and we shall in the main confine ourselves to the properties of air, though the mechanical results obtained are applicable in general to gases and vapours. The science which relates to the mechanical properties of the air is called Pneumatics. It belongs to Chemistry to treat of the special properties of each gas.

XLII. AIR A SUBSTANCE.

482. The atmosphere is a thin fluid which surrounds the globe, and is necessary for the support both of animal and vegetable life. Although before attention has been drawn to its properties it might be imagined that air is scarcely a form of matter, yet on due consideration it will be found to be such, though in a very rarefied condition.

483. The air is generally supposed to be transparent, but when we look at a cloudless sky we recognise a blue colour which may be attributed to the air. The fact that this colour is not visible when we inspect a small quantity of air by itself is consistent with other facts of a similar kind. Thus sea-water in a large mass presents a greenish tint, but a small quantity of it seems without colour. So also wine in a very slender glass appears much paler in tint than in a wider glass.