PNEUMATICS visible relative motion set up among the parts of a fluid rapidly decays in virtue of viscosity, which even for the subtlest gases is quite appreciable in its effects. In a fluid at rest, then, the pressure over any surface which we may imagine to be drawn is perpendicular (or normal) to the surface at every point of it ; and from this it follows at once, as has been proved in HYDROMECHANICS, vol. xii. p. 439, that the pressure at any point of a fluid at rest has the same value in all possible directions. ,e The pressures at two contiguous points in a fluid may urces. either differ or not. If they differ, the change of pressure must be balanced by some extraneous force acting on the fluid in the direction in which the pressure increases. Any direction in which no such force acts must be a direction in which there is no change of pressure ; otherwise, equi librium will be destroyed. Suppose now the resultant force at every point in a fluid at rest to be given. In directions at right angles to the force at any given point the pressure will not vary. Hence we can pass to an infinite number of contiguous points at which the pressure is the same as at the given point. By making each of these in turn the starting-point, we can pass on to another set of points, and so gradually trace out within the fluid a surface at every point of which the pressure is the same. Such a surface is called a surface of equal pressure, or briefly a level surface ; and we see from the mode of its construc tion that it is at every point of it perpendicular to the resultant force at that point. Imagine any two contiguous level surfaces to be drawn, at every point of the one of which the pressure is p, at every point of the other p + ftp. Consider the equilibrium of a small column of average density p, bounded at its ends by these surfaces. Let A be the area of each end, and or the length of the column or perpendicular distance between the level surfaces. If R is the average resultant force per unit mass acting on the column, then we have, for equilibrium of the column, or the rate of increase of the pressure at any point per unit of length at right angles to the level surface is equal to the resultant force per unit of volume at that point. If the applied forces belong to a conservative system, for which V is the potential (see MECHANICS), we may write the equation in the form Sp = -pSV. Eq po- Hence over any equipotential surface, for which 8V = 0, p uiiul i s constant, and is therefore a function of V. Consequently es> p also is a function of V. For a fluid in equilibrium, therefore, and under the influence of a conservative system of forces, the pressure and density are constant over every equipotential surface, that is, over every surface cutting the lines of force at right angles. Now in the case of gases, to which our attention is at present confined, the density (temperature remaining con stant) varies with every change of pressure; in mathemat ical language p is a function of p. Thus, before we can solve the equation of equilibrium for a gas, we must be able to express this function mathematically ; in other words, we must know the exact relation between the density of a gas and the pressure to which it is subject. This problem, which can only be settled by experiment, was solved for the case of air within a certain range of pressures by Robert Boyle (1662). Before discussing his results and the later results of other investigators, we shall first consider the general properties of our atmo sphere as recognized before Boyle s day. It is evident that, for a fluid situated as our atmosphere is, the pressure must diminish as we ascend. The equi potential surfaces and consequently the surfaces of equul pressure and of equal density will be approximately spheres concentric with the earth. At any point there will be a definite atmospheric pressure, which is equal numerically to the weight of the superincumbent vertical column of air of unit cross-section. The effect of this pressure, as exemplified in the action of the common suction-pump, seems to have been first truly recognized by Galileo, who showed that the maximum depth from which water can be pumped is equal to the height of the water column which would exert at its base a pressure equal to the atmospheric pressure. As an experimental verification, he suggested filling with water a long pipe closed at the upper end, and immersing it with its lower and open end in a reservoir of the same liquid. The liquid surface in the pipe would, if the pipe were long enough, stand at a definite height, which would be the same for all longer lengths of pipe. The practical difficulty of constructing a long enough tube (33 feet at least) prevented the experi ment being really made till many years later. Torricelli, however, in 1642, by substituting mercury for Torri- water, produced the experiment on a manageable scale. As ce Ui 8 ex- mercury is denser than water in the ratio of about 13.6:1, P enmeut - the mercury column necessary to balance by its weight the atmospheric pressure will be less than the water column in the inverse ratio, or a little under 30 inches. Torricelli s experiment is exhibited in every mercurial barometer (see BAROMETER and METEOROLOGY). By this experiment he not only gave the complete experimental verification of Galileo s views relating to atmospheric pressure, but pro vided a ready means of measuring that pressure. The most obvious applications of the barometer are these : (1) to measure the variation in time of atmo spheric pressure at any one locality on the earth s sur face (the existence of this variation was discovered soon after the date of Torricelli s experiment by Pascal, Descartes, Boyle, and others) ; (2) to measure the varia tion of atmospheric pressure with change of height above the earth s surface (Descartes mentions this application in the Principia Philosophic, 1644; but to Pascal is the honour due of having first carried the experiment into execution, 1647) ; and (3) to compare pressures at different localities which are on the same level (if the pressures are equal, the air is in equilibrium ; if they are not, there must be flow of air from the place of higher pressure to that of lower in other words, there must be wind, whose direction of motion depends on the relative position of the places, and whose intensity depends on the distance between the places and the difference of pressures). The first and last of these measurements are of the greatest importance in meteorology. The second is a valuable method for measur ing attainable heights, and is intimately connected with the problem as to the relation between the pressure and density of the air. Thus it would be possible, by barometric observations at a series of points in the same vertical line, to obtain a knowledge of this relation more and more truly approximate the closer and more numerous the points of observation taken. At best, however, such a method could give the law connecting density with pressure for those pressures only which are less than the normal atmo spheric pressure. The problem is better solved otherwise. Assuming Boyle s law that the density of air is directly as the Relation pressure, we cau now integrate the equation of equilibrium between height and put it in the form where p is the pressure at zero potential and K is the constant ratio of the pressure to the density. For all attainable heights in our atmosphere we may assume the force of gravity to be the same. Hence ve may write V = </A., XIX. 31 pressure in our at
mosphere.