the places at which the above result for the barometric pressure at the level of the ocean was found, we shall have 51°,9 as answering to the temperature of the air at the lower station. Hence,

48,851,9

2

30,55 20 nearly; and 20 × 0,00223 = 0,45.

The correction, therefore, for difference of temperature, is

which added to 33 gives 34,5 for the elevation of the place of observation at Cambridge above the level of the sea. Now the actual elevation of the cistern of the barometer, as carefully ascertained by levelling, is found to be 31 feet. In the calculation of very small heights near the level of the ocean, it is very common to dispense with the formula and adopt the following rule, namely, as 0,1 is to the difference in the barometric columns, so is 87 feet to the approximate difference of level required; which is to be corrected, if necessary, for the difference from 31° of the mean temperature of the air at the two stations. Thus,

0,1 0,038 87: 33,1,

a result agreeing very nearly with that derived from the formula.

Thus, under a pressure of 30 inches of mercury at the temperature of 50°, 0,1 of an inch of mercury answers to 87 feet of atmosphere. It will be seen moreover, that, as 0,1 of an inch of mercury is equivalent to 87 feet of air, 0,01 answers to 8,7, 0,001 to 0,87, and to 1,14. Hence in a good mountain barometer, graduated to 500dths of an inch, there will be a sensible difference in the pressure of the air arising from a change of altitude of less than two feet, or two thirds the length of the instrument.

Τ

Formula (IV.) is essentially the same with that given by Laplace in the 10th book of the Mécanique Céleste, but simplified after the example of Poisson, and reduced to English measures. The following example will serve to illustrate every part of this formula.

At the lower of two stations, the mercury in the barometer was observed to be 29,4 inches, and its temperature 50°, that of

the air being 45°; and at the upper station, the height of the barometer was 25,19, its temperature 46°, and that of the air 39o, the latitude of the place being 30°.

Laplace's formula, applied to the same example, gives 688,97 fathoms, differing from the above only 1,22; whereas, by Sir George Shuckburgh's method, in which no account is taken of the variation of gravity, either for difference of latitude or difference of elevation in the same latitude, the result is 685,125. This corresponds with the approximate height derived from the first correction in the above example.

475. We have already mentioned, that, unless very particu lar precautions are taken, mercury is depressed in glass tubes, and that this depression is inversely proportional to the diame ter of the tube. It is always indicated, moreover, when it takes place by the upper surface being convex. It is not necessary to have regard to this circumstance in the calculation of heights. by the barometer, where the two observations are taken with the same instrument, since the difference in the length of the ba rometric columns would be the same, whether they were corrected or not. † But in order that observations by different instruments, liable to different capillary effects, may be strictly compared with each other, a correction should be applied, which may be readily done by means of the following table.

† Also in a syphon barometer, or one in which the tube, instead of entering a basin, turns up at the bottom and continues of the same bore, as in figures 230, 232, since the capillary effect is the same in both branches, the observed altitude reckoned from the surface in the shorter branch, would not be affected by the correction.

HYDRODYNAMICS.

Of the Discharge of Fluids through Apertures in the Bottom and
Sides of Vessels.

476. If a fluid be made to pass through a canal or tube of variable bore, kept constantly full, and the velocity be the same in every part of the same section, since for any given time the same quantity of fluid must pass through every section, this quantity must be equal to the area of the section multiplied by the velocity. 6, 6', being the areas of two sections, and v, v', the velocities at these sections, we shall have

that is, the velocities in different sections are inversely as the areas of the sections.

The case here supposed is purely theoretical, and can never occur in practice, since on account of friction, the velocity is always greatest at the surface in a canal, and at the axis in a tube.

477. Let MNOP represent a vessel filled with a fluid up Fig.233. to GH, CD an aperture, very small compared with the bottom MP, CIKD, the column of fluid directly above the aperture, and CABD the lowest lamina or stratum of this fluid, immediately contiguous to the aperture. Also let v denote the velocity acquired by a heavy body in falling freely through BD, the height of the stratum, and u the velocity which the same stratum would Mech.

47

acquire in falling through the same space by the pressure of the column CIKD. If we suppose the lowest stratum ACDB, to fall as a heavy body through the height BD, the moving force will be its own weight. But if we suppose it to be urged by its own weight, together with the pressure of the incumbent column. of fluid CIKD through the same space, the velocity in the former case will be to that in the latter, as the moving forces and the times in which they act, the mass moved being the same in both cases. But the moving forces are to each other, as the heights BD, KD, and the times in which they act, the space 264. being the same, are inversely as the velocities. Accordingly,

27.

Now is the velocity which a heavy body would actually acquire in falling through the space BD, and as the velocities, other things being the same, are as the square roots of the spa268. ces, u the velocity of the issuing fluid is that which a heavy body would acquire in falling through KD, the height of the fluid above the orifice. Therefore, the velocity with which a fluid is discharged from the bottom of a vessel is equal to that acquired by a heavy body in falling through a space equal to the height of the fluid above the orifice. Also if a pipe A'B'C'D' be inserted horizontally, or inclined in any way to the horizon, it may be shown, in like manner, since the pressure of fluids is equal in all directions, that the fluid will be discharged with the same velocity as before. It will accordingly ascend to the level of the fluid in the vessel, all obstructions being removed; and it is found in fact, under the most favourable circumstances, nearly to reach this point. It follows, moreover, from what is above laid down, that if apertures be made at different distances s, s', s", below the surface, the velocities at these points, and consequently the quantities of fluid discharged at these points, from aperture's of