above, the resulting value of ad will be a first approximation to the required height. By experiments it has been found that the relative height thus obtained varies by 3 of its value for each degree of the thermometer in the difference between 31° and the mean of the temperature of the air at the two stations: consequently, if d be the difference between 31° and the mean of two detached thermometers, one at each station, the correction on account of the temperature of the ad air will be expressed by the formula d, where ad is the 435 first approximate value of the height. This correction is to be added to that approximate height when the mean of the detached thermometers is greater than 31°, and subtracted when less. : The result is very near the truth when the height of one station above the other does not exceed 5000 or 6000 feet and when the difference of temperature does not exceed 15 or 20 degrees in other cases, more accurate formulæ must be employed, and that which is given by Poisson in his " Traité de Mécanique" (second ed. No. 628.), when the measures are reduced to English yards, and the temperatures to those which would be indicated by Fahrenheit's thermometer, is H = A { log. barom. at a log. k} ; in which H is, in yards, the required height of one station above the other, as d above a. t+t' - 64° T T k = height of barom. at d, x (1 + 9990) t and t' are the temperatures of the air, by detached thermometers at the two stations. T and T' are the temperatures of the mercury, by attached thermometers. A is the common latitude of the stations, or a mean of the latitudes of both if the stations be distant from each other in latitude. 431. The siphon barometer is a glass tube formed nearly as in the annexed figure, and containing mercury; it is hermetically sealed at both extremities, and has at A a very fine perforation, which allows a communication with the external air without suffering the mercury to escape. The atmosphere pressing on the mercury at N, balances the weight of a column of that fluid whose upper extremity may be at M. There is M a sliding vernier at each extremity of the column; and, the zero of the scale of inches being below N, the difference between the readings at M and N on the scale is the required height of the column of mercury. M In other mountain barometers the tube is straight, and its lower extremity, which is open, enters into a cistern AB containing mercury: the bottom of the cistern is of leather, and by means of a screw at C, that mercury can be raised or lowered till its upper surface passes through an imaginary line, on which, as at N, is the zero of the scale of inches; M being the upper extremity of the column of mercury in the tube, the height MN is read by means of a vernier at M. The external air presses on the flexible ^ bottom of the cistern, and this causing the surface N of the mercury at N to rise, or allowing it to fall, the corresponding variations in the elasticity of the air in the part AN of the cistern, produce the same effect on the height of the mercurial column MN, as would be produced by the external air if it acted directly on the surface at N. The barometer invented by Sir H. Englefield has no screw for regulating the surface of the mercury in the cistern with respect to the zero of the scale of inches; and the atmosphere, entering through the pores of the box-wood of which the cistern is formed, presses directly at N on the surface of the mercury; there can, consequently, be only one state of the atmosphere in which the surface is coincident with that zero. The exact number of inches and decimals, on the scale, at which the extremity м of the column of mercury stands when the surface at N coincides with the zero is found by the artist, and engraven on the instrument; and, when the top of the column is at that height (or at the neutral point, as it is called) no correction is necessary on account of the level of the mercury in the cistern. In other cases such correction is determined in the following manner: The ratio between the interior area of a horizontal section through the cistern, and the area of a like section through the bore of the tube, is ascertained by the artist and engraven on the instrument: let this ratio be as 60 to 1: then the lengths of cylindrical columns, containing equal volumes, being inversely proportional to the areas of the transverse sections, the required correction will be of the difference between the height of the neutral point, and that at which the top of the column stands in the tube. This correction must be added to, or subtracted from, the height read on the scale according as the top of the column is above or below the neutral point. of The difficulty of transporting the usual mountain barometer overland has induced travellers to use, for the purpose determining the relative heights of stations, the instrument invented by Dr. Wollaston, and called by him a thermometrical barometer. This consists of a thermometer, which may be of the usual kind, but very delicate, whose bulb is placed in the steam arising from distilled water in a cylindrical vessel about five inches long, the water being made to boil by an oil or spirit lamp. Now it is known that water boils when the elastic power of the steam produced from it is equal to the incumbent pressure of the atmosphere; and thus, the temperature at which, in the open air, the water boils, will depend upon the weight of the atmospherical column above it. Therefore, since this weight becomes less as the station is more elevated, it is evident that water will boil at a lower temperature on a mountain than on a plain at its foot; and, for the purpose of determining the height of the mountain, it is only necessary to find an expression for the elastic power of steam, at a given temperature under the pressure of the atmosphere, in terms of the height of an equivalent column of mercury in a barometer. From the experiments of De Luc, M. Dubuat has obtained a formula equivalent to or log. h = 3.84 (log. t — 1.876105), or log. h = where h is the height of the ordinary barometrical column expressed in English inches; t is the temperature at which the water boils in the open air, expressed by degrees of Fahrenheit's thermometer, and reckoned from the freezing point (32°) as zero. The value of h being thus obtained for each of the two stations, the height of one station above the other may be found by the formula for the usual mountain baro meter. 432. The processes of a trigonometrical survey for determining the figure of the earth are too extensive to be frequently put in practice; they also require the combined efforts of many persons, and they involve expenses which are beyond the resources of private individuals. On the other hand, the variations of terrestrial gravity at different places on the earth's surface are capable of being determined with great precision by comparing together the observed times of the vibrations of pendulums at the places: and as these variations, though in part due to a want of homogeneity in the mass of the earth, depend chiefly on the deviation of its figure from that of an exact sphere; a series of well-conducted experiments on the vibrations of pendulums at stations remote from each other, which may be made by two or three persons at a comparatively small expense, afford the most convenient means of ascertaining the form of the earth. 433. The pendulums which are employed for this purpose are generally of the kind called invariable; that is, they are not provided with a screw by which their lengths may be increased or diminished like the pendulums applied to ordinary clocks; so that their lengths can only vary by the expansion or contraction of the metal in consequence of the variations of temperature to which they may be subject. The effects arising from the variations of temperature and from all the other circumstances which interfere with the action of gravity upon them, are determined by theory and applied as corrections to the observed times of the vibrations. These pendulums are employed in two ways: they may be attached to the machinery of a clock for the purpose of continuing the oscillations and registering their number, or they may be unconnected with any maintaining power, and left to vibrate by the action of gravity till the resistance of the air and the friction on the point of suspension bring them to a state of rest. The late Captain Kater, availing himself of that property of vibrating bodies by which the centres of suspension and oscillation are convertible, constructed pendulums which admit of being made to vibrate upon either of those centres at pleasure; by this construction the effective length of the pendulum (which is the distance between those centres) is easily found, by measurement, when the places of the centres have been determined by the experimental number of vibrations made upon each being equal, in equal times; and such invariable pendulums are now generally employed by the English philosophers for the purposes of experiment. They are furnished with steel pivots or axles (called knife edges) at the two places which are to be made alternately the centres of suspension and oscillation; and these rest upon the upper edge of a prism of agate or wootz, so that the pendulums may vibrate with as little friction as possible. 434. In making the experiments with a detached pendulum, the latter is placed in front of a clock regulated by mean solar, or sidereal time, but quite unconnected with its motion. On the pendulum of the clock is placed a disk of white paper opposite to a vertical wire crossing an opening in the rod of the detached pendulum. A telescope is fixed a few feet in front of the pendulums, so that, when these are put in motion, the disk of paper may be seen to pass over the field of view. Now, since the detached pendulum and that of the clock have not exactly the same velocity, if we suppose the vertical axis of the former to have been originally in coincidence with the centre of the disk, these will separate from each other by the excess of the velocity of one above that of the other: but after a certain number of oscillations, they will again coincide, moving in opposite directions; and then the detached pendulum may be said to have gained or lost one oscillation. For example, if 30 vibrations of the clock pendulum had been observed in the interval between two consecutive coincidences, it is evident that the invariable pendulum must have made either 29 or 31 vibrations. After being in coincidence the pendulums separate as before, and again, subsequently, they coincide; and so on. The number of oscillations made by the pendulum of the clock between two, three, or more coincidences is counted, and the times of the several coincidences are shown by the clock, when they take place: then the number of oscillations made by the clock pendulum in the time of any number of coincidences (suppose n) is to the number of oscillations of the detached pendulum in the same time (which number in this case will exceed or fall short of the former number by n), as 86400 seconds (the number of oscillations made by the clock in a mean solar, or sidereal day, according as the clock is regulated by solar or sidereal time) are to the number of oscillations made by the detached pendulum in the same time, at the station. With respect to the invariable and attached pendulum, the number of vibrations performed by it in a mean solar, or sidereal, day is ascertained by observing the times indicated on the dial of the clock to which it is attached, at the end of equal intervals of time, as 12 or 24 hours. And in both cases it is most convenient to determine the measure of time by the transits of stars. Captain (now Colonel) Sabine's experiments in his two voyages during the years 1822, 1823, were made with pendulums whose lengths were invariable except in respect of temperature. In the first voyage the pendulum was detached from any clock-work, so that after a certain number of oscillations it rested: but in the second voyage the pendulum was attached to a clock, and its oscillations were continued by the maintaining power of the clock. The uniformity of the maintaining power was inferred from that of the extent of the arc of vibration. 435. The following propositions contain the principal subjects relating to the corrections which are required for the |