1 would be in equilibrium if B the space BC. This supposes that the surface AB and the orifice at Care exposed to the same pressure, as for instance that of the atmosphere, which will be explained hereafter. If the pressure at the level AB is greater than at C, the effect is the same as if the height BC were increased to the extent which would correspond to this excess of pressure; and similarly if the pressure at the level AB is less than at C the height BC must be supposed diminished to a corresponding extent. Each particle of liquid on leaving the vessel will describe a parabola by virtue of the principles of Chapter XX.; and thus by the continuous stream of particles we obtain a visible representation of the parabolic course. 466. If we suppose the hole to be in the shape of a horizontal pipe the liquid will issue in a horizontal direction, so that the particles start from the highest point of their course and afterwards continually descend. But we may if we please insert at Ca short pipe inclined to the horizon upwards, and then the fluid will ascend obliquely to some height before it begins to descend. Or the short pipe may be first horizontal for a brief space, and then turn vertically upwards: in this case the liquid spouts vertically upwards, and according to theory would rise to the level of AB. 467. Although the theory on which the preceding Article depends is beyond the range of the present work, yet there is one part of the result involved in it of which the reasonableness may be rendered tolerably evident; and this process well deserves attention. The liquid at C issues with a certain velocity, namely, with that which would be acquired in falling freely down BC. Hence if we want the liquid to issue with twice this velocity we must make the hole, not at twice the depth of C below the surface, but at four times this depth: that is, we have as it were to provide four times the pressure in order to secure twice the velocity. But the apparent difficulty is 'soon removed. For since the velocity at the lower hole is to be double that at the higher hole, each particle issues from the lower hole with double the velocity with which it issues from the higher hole; and moreover supposing the holes to be of the same size, double the number of particles will issue in the same time from the lower hole as from the higher hole. Thus, in all, we have at the lower hole four times the effect produced which is produced at the higher hole, corresponding, as might be expected, to the circumstance that the pressure at the lower hole in equilibrium is four times that at the higher. 468. There are two cases of the motion considered in Art. 466, namely, that in which the liquid in the vessel is always maintained at the same level, and that in which it is not. In the latter case the value which theory gives for the velocity does not agree with observation when the level of the descending fluid comes near the hole. But in both cases, so long as the hole is not too near the surface of the liquid the actual velocity of the issuing liquid does not differ much from the value assigned by theory. But when the liquid is made to spout vertically upwards it does not reach the level of the liquid in the vessel; the velocity of the issuing fluid is diminished by the friction against the sides of the pipe or opening through which it escapes, and the resistance of the air also produces a sensible effect. 469. If we know the size of a hole and the velocity with which liquid is escaping through it, we can calculate 'the amount of liquid which will flow out in an assigned time. But in making such calculations and comparing the results with observation it is found that the theoretical estimate is too large. Some curious phenomena have been noticed in connexion with this subject. We will suppose that the hole is very small, that it is in the base of the vessel, and that the base is very thin; this special 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 explana tion. 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, cohesion 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. |