let this be p. Then the volume v of the mercury is given by the equation

W

and this volume is equal to the product of the area a of the cross-section and the length of the tube. Hence

V W
A===

If the length be measured in centimetres and the weight grammes, the density being expressed in terms of grammes per c.c., the area will be given in sq. cm.

The length of the mercury column is not exactly the length of the tube, in consequence of the fingers closing the tube pressing slightly into it, but the error due to this cause is very small indeed.

This gives the mean area of the cross-section, and we may often wish to determine whether or not the area of the section is uniform throughout the length. To do this, carefully clean and dry the tube as before, and, by partly immersing in the trough, introduce a thread of mercury of any convenient length, say about 5 centimetres long. Place the tube along a millimetre scale, and fix it horizontally so that the tube can be seen in a telescope placed about six or eight feet off.

By slightly inclining the tube and scale, adjust the thread so that one end of it is as close as possible to the end of the tube, and read its length in the telescope. Displace the thread through 5 cm. and read its length again; and so on, until the thread has travelled the whole length of the tube, taking care that no globules of mercury are left behind. Let 1, 12, 13.... be the successive lengths of the thread. Then run out the mercury into a beaker, and weigh as before. Let the weight be w, and the density of the mercury be p.

Then the mean sectional areas of the different portions of the tube are

The mean of all these values of the area should give the mean value of the area as determined above. The accuracy of the measurements may thus be tested.

On a piece of millimetre sectional paper of the same length as the tube mark along one line the different points which correspond to the middle points of the thread in its different positions, and along the perpendicular lines through these points mark off lengths representing the corresponding areas of the section, using a scale large enough to shew clearly the variations of area at different parts of the length. Join these points by straight lines. Then, the ordinates of the curve to which these straight lines approximate give the cross-section of the tube at any point of its length.

Experiment.-Calibrate, and determine the mean area of the given tube.

Enter the result thus :

[The results of the calibration are completely expressed by the diagram.]

Mean of the five determinations for calibration 0'409 sq. mm.

MEASUREMENT OF VOLUMES.

The volumes of some bodies of known shape may be determined by direct calculation from their linear dimensions; one instance of this has been given in the experiment with the calipers.

A Table giving the relations between the volume and linear dimensions in those cases which are likely to occur most frequently will be found in Lupton's Tables, p. 7.

9. Determination of Volumes by Weighing.

Volumes are, however, generally determined from a knowledge of the mass of the body and the density of the material of which it is composed. Defining 'density' as the mass of the unit of volume of a substance, the relation between the mass, volume and density of a body is expressed by the equation M=Vp, where M is its mass, v its volume, and p its density. The mass is determined by means of the balance (see p. 123), and the density, which is different at different temperatures, by one or other of the methods described below (see pp. 139-143). The densities of certain substances of definitely known composition, such as distilled water and mercury, have been very accurately determined, and are given in the tables (Nos. 32, 33), and need not therefore be determined afresh on every special occasion. Thus, if we wish, for instance, to measure the volume of the interior of a vessel, it is sufficient to determine the amount and the temperature of the water or mercury which exactly fills it. This amount may be determined by weighing the vessel full and empty, or if the vessel be so large that this is not practicable, fill it with water, and run the water off in successive portions into a previously counterpoised flask, holding about a litre, and weigh the flask thus filled. Care must be taken to dry the flask between the successive fillings; this may be rapidly and easily done by using a hot clean cloth. The capacity of vessels of very considerable

size may be determined in this way with very great accuracy.

All the specific gravity experiments detailed below involve the measurement of a volume by this method.

Experiment.-Determine the volume of the given vessel.
Enter results thus :-

10. Testing the Accuracy of the Graduation of a Burette.

Suppose the burette to contain 100 c.c. ; we will suppose also that it is required to test the capacity of each fifth of the whole.

The most accurate method of reading the burette is by means of a float, which consists of a short tube of glass loaded at one end so as just to float vertically in the liquid in the burette; round the middle of the float a line is drawn, and the change of the level of the liquid is determined by reading the position of this line on the graduations of the burette. The method of testing is then as follows:

Fill the burette with water, and read the position of the line on the float. Carefully dry and weigh a beaker, and then run into it from the burette about th of the whole contents; read the position of the float again, and weigh the amount of water run out into the beaker. Let the number of scale divisions of the burette be 2012 and the weight in grammes 20119. Read the temperature of the water; then, knowing the density of water at that temperature (from table 32), and that I gramme of water at 4° C. occupies I c.c.,

we can determine the actual volume of the water corresponding to the 2012 c.c. as indicated by the burette, and hence determine the error of the burette. Proceeding in this way for each th of the whole volume, form a table of corrections.

Experiment.-Form a table of corrections for the given burette. Enter results thus :

The angle between two straight lines drawn on a sheet of paper may be roughly measured by means of a protractor, a circle or semi-circle with its rim divided into degrees. Its centre is marked, and can therefore be placed so as to coincide with the point of intersection of the two straight lines; the angle between them can then be read off on the graduations along the rim of the protractor. An analogous method of measuring angles is employed in the case of a compassneedle such as that required for § 69. Angles traced on a diagram may be determined by measuring lines from which one or other of the trigonometrical ratios can be calculated (see Chap. V.*).

The more accurate methods of measuring angles depend on optical principles, and their consideration is accordingly. deferred until the use of the optical instruments is explained (see §§ 62, 71).

MEASUREMENT OF SOLID ANGLES.

The angle which a plane curve joining any two points subtends at a third point o in the plane of the curve, as given by its 'circular measure,' may be found thus: