within the limits, there is a definite point on the stem to which the instrument will sink, depending on the specific gravity; and the stem can be graduated in such a manner that the graduation reading gives the specific gravity at once. This is generally done by a scale attached to the inside of the stem, and hence all that has to be done to determine the specific gravity of a liquid is to float in it a suitable hydrometer, and take the scale reading at the surface. The temperature correction is to be allowed for as usual.

An instrument sensitive to such slight variations of density as o' per cent. would require to have too long a stem if used for the whole range of density commonly occurring. Hydrometers are, therefore, usually obtained in sets of three or four, each suitable for one portion only of the range. The case in which they are kept contains a long cylindrical vessel, which is convenient for floating them in and also a thermometer.

The hydrometers, vessel, and thermometer should be carefully washed and dried before replacing them in the

case.

The graduation of the scale is a comparatively difficult matter, as equal increments in the length of the stem immersed do not correspond to equal differences of density. The scales are graduated by the instrument-makers, and we require to be able to test the accuracy of the graduation.

We can do this by taking the hydrometer readings in liquids whose specific gravities are known. Distilled water would naturally be a suitable one for the purpose. The hydrometer when floating in distilled water at 15°C. should read o'999. The specific gravity of any other suitable liquid could be determined by one of the methods already described. The following experiment, however, serves as a very instructive method of comparing the density of any liquid with that of water, and it is, therefore, suggested as a means of testing the accuracy of the hydrometer scale.

To compare the Densities of two Liquids by the Aid of the Kathetometer.

FIG. II.

If we have a U tube (fig. 11) and fill one leg with one liquid standing up to the level P, and the other with a second up to the level Q, and if R be the common surface of the liquids in the two legs PR, QR, their densities are inversely proportional to the vertical distances between P and R, Q and R.' These can be accurately measured by the kathetometer, and the densities thus compared. If the kathetometer be not available, the heights may be measured by scales placed behind the tubes, which are read by a telescope placed at a distance and roughly levelled for each observation.

R.

This arrangement supposes that the two liquids do not mix. The following apparatus is therefore more generally available:

A B C, D E F are two U tubes, the legs BC, D E being the shorter. These legs are connected together by a piece of india-rubber tubing C G D.

One liquid is poured into the tube

A B, and then the other into the tube F E. This, as it runs down the tube, compresses the air below it, thus increasing the pressure on the surface of the first liquid, and forcing it up the leg BA. The quantity poured into FE must not be sufficient to rise over the end D of the tube.

F

FIG. 12.

G

E

B

Now pour more of the first liquid into A B. This forces up the level of the liquid in EF, and after one or two repetitions of this

See below, chap. vii. p. 197.

operation the levels of the liquid in one tube will be at A and c, those in the other being at F and D.

The pressure at C and D, being that of the enclosed air, is the same.

The excess of the pressure at c above the atmospheric pressure is due to a column of liquid of height equal to the vertical distance between A and C, that at D is due to a column of the second liquid of height equal to the distance between F and D.

These distances can be observed by the kathetometer, and the densities of the two liquids are inversely proportional to them.

The surface of the liquids in the tubes will be curved, owing to capillary action. In measuring, either the bottom or the top of the meniscus, whichever be most convenient, may be observed, but it is necessary to take the same at each end of the column. The bottom will, if the liquid wet the tube, give the more accurate result.

It is well to hang up behind the tubes a sheet of white or grey paper, to afford a good background against which to see the liquids.

It is important that the temperature should remain the same during the experiment; for if it increase the pressure in the portion CGD increases, and the air there expands, thus forcing up the columns of liquid. We may avoid the difficulty this causes by the following method of taking the measurements:

Observe the height of A, then the height of c, and finally the height of A again.

Then, if the temperature has changed uniformly and the intervals between the successive measurements have been the same, the mean of the two observed heights of A will give its height at the time when the observation of the height of c was made, and the difference between these two, the mean of the observed heights of A and the height of c, will give the true height of the column.

If one liquid be water at a temperature, say, of 15° C., the ratio of the two heights gives us the specific gravity of the second liquid, for its temperature at the time of the observation, referred to water at 15° C.

If we wish to find the true specific gravity of the liquid at the temperature of the observation, 15° C., we must multiply the above ratio by the specific gravity of water at 15° C.

Suppose the second liquid is also at 15° C., and that its coefficient of expansion by heat does not differ greatly from that of water. Then the same ratio gives us the specific gravity of the liquid at 4° C. referred to water at 4° C., or the true specific gravity of the liquid at 4° C. without any correction.

Experiment.-Determine the specific gravity of the given liquid by means of the hydrometer, testing the accuracy of the results.

CHAPTER V.*

MEASUREMENT OF VELOCITY AND ACCELERATION.

B. To Measure the Velocity of a Pendulum. WHEN a body is moving uniformly in a straight line its velocity is measured by measuring the distance it traverses in a measured interval. When the velocity is changing, this method is no longer applicable. The measurement of the distance traversed in a known interval gives only the mean velocity for that interval, but by making the interval sufficiently short the result represents adequately the actual

FIG. vi.

B

velocity during the interval. We shall in this chapter shew how by making use of the very short intervals corresponding to the time of vibration of a tuning-fork a fair measure of the velocity of a moving body can be obtained, and shall further shew how by geometrical methods upon a true scale-diagram of the path of a moving body the velocity and acceleration of the body can be determined. The velocity of the pendulum is not uniform at any

point of its path; but when near its lowest position it has a maximum value which varies very slowly. This maximum value may be found thus in the case of a heavy pendulum