fix ideas on the subject, consider the determination of the mass of a given volume of gas contained in a glass globe, by weighing the globe full and empty. During the interval between the two weighings the temperature and pressure of the air, and in consequence the apparent weight of the glass vessel, may have altered. This change, unless allowed for, will appear, when the subtraction has been performed, as an error of the same actual magnitude in the mass of the gas, and may be a very large fraction of the observed mass of the gas, so that we must here take account of the variation in the correction for weighing in air, although such a precaution might be quite unnecessary if we simply wished to determine the actual mass of the glass vessel and its contents to the degree of accuracy that we have hitherto assumed. A case of the same kind occurs in the determination of the quantity of moisture in the air by means of drying tubes (§ 42).

Cases of the second kind referred to above often arise from the fact that the formulæ contain differences of nearly equal quantities; we may refer to the formula employed in the correction of the first observations with Atwood's machine (§ 21), the determination of the latent heat of steam (§ 39), and the determination of the focal length of a concave lens (§ 54) as instances. In illustration of this point we may give the following question, in which the hypothetical errors introduced are not really very exaggerated.

'An observer, in making experiments to determine the focal length of a concave lens, measures the focal length of the auxiliary lens as 10.5 cm., when it is really 10 cm., and the focal length of the combination as 145 cm., when it is really 15 cm. ; find the error in the result introduced by the inaccuracies in the measurements'

or more than 25 per cent., whereas the error in either observation was not greater than 5 per cent.

It will be seen that the large increase in the percentage error is due to the fact that the difference in the errors in F and f has to be estimated as a fraction of F-f; this should lead us to select such a value of fi as will make F-f1 as great as possible, in order that errors of given actual magnitude in the observations may produce in the result a fractional error as small as possible.

We have not space for more detail on this subject. The student will, we hope, be able to understand from the instances given that a large amount of valuable information as to the suitability of particular methods, and the selection of proper apparatus for making certain measurements, can be obtained from a consideration of the formulæ of reduction in the manner we have here briefly indicated.

Graphical Methods.

The results of a large number of experiments can be best expressed graphically. Examples of this method will be found in the course of the book. (See specially §§ 26, 40 41.)

E

The method is chiefly useful in cases in which we wish to trace the dependence of one quantity on another. Paper suitable for the purpose, ruled in small squares, can be easily obtained.'

In applying the method, the values of the independent variable are set down as abscissæ parallel to one set of lines, the corresponding values of the dependent variable being measured as ordinates at right angles to this. In cases in which the phenomenon under investigation is continuous in its character, a smooth curve can usually be drawn, either freehand or by the aid of a flexible ruler, so as to pass approximately through these points, and the law sought can be obtained by an investigation of the form of the

curve.

Thus, suppose we are endeavouring to prove that the pressure of a given mass of gas at constant volume varies as the absolute temperature, we lay off as abscissæ the observed values of the temperature, say in degrees centigrade from freezing point as zero, and as ordinates the corresponding pressures.

On drawing the curve which best represents the experiments we find it to be a straight line; moreover, this line cuts the line of no pressure from which the ordinates are measured at a point on the negative side of the origin about 273° C. below freezing point. This point is the absolute zero, and the pressure is clearly proportional to the temperature reckoned from it.

Let us

The accuracy of a result obtained by a graphical method will, to some extent, depend on the scale adopted. suppose that in the above experiment we can read the temperature to o'1°C., and the pressure to 5 mm. Then it is clear we must adopt such a scale for the temperature, if we wish to be accurate, as will allow o'r°C.

Messrs. Waterlow supply paper ruled in inch squares. Each inch is subdivided to tenths by fine lines, the half-inch lines being thicker than the others. For some remarks on different 'squared' papers see p. 11 of the Report on Spectrum Analysis, B. A. Report, 1881.

to be clearly visible. We might take 1 inch to repre

sent 1°.

If at the same time we represent 1 cm. of pressure by I inch on the diagram, we can plot down the pressure to 5 mm., and these scales will give us satisfactory results. The figure so drawn will be very large, larger than is required for the accuracy attempted in most of the experiments described.

When the diagram is to be used to represent the variations of one quantity corresponding to those of another over a small range, a wide scale can be used without making a very large diagram by using the abscissæ or ordinates, or both, to represent the respective changes and not the whole quantities. Thus, suppose we wish to represent the changes. of volume of one gramme of water consequent on changes of temperature between o° C. and 10° C.; we may regard the horizontal line through the origin as indicating volumes equal to that of one gramme of water at 4° C., and one inch of vertical height may represent a change of volume of '00001 C.C. The line of no volume would, if drawn, be 100,000 inches below the horizontal through the origin. But it need not be drawn ; and if one inch of horizontal distance represent 1° C., the whole diagram will be comprised in a space 10 inches square.

In drawing a diagram the horizontal and vertical scales chosen should always be very clearly set out in the diagram itself.

The Slide Rule.

The slide rule is a mechanical contrivance for performing rapidly various arithmetical operations. Its action depends in the main on the two principles that the logarithm of the product of two numbers is the sum of the logarithms of its factors, and that the logarithm of the nth power of a number is n times the logarithm of the number.

In its very simplest form a slide rule would consist of two identical scales, one of which can slide along the other. The scales are divided in such a way that the distance along either scale measured from one end-say, the lefthand-is proportional to the logarithm of the corresponding scale number. Thus the distance from the left-hand end to a reading a, say, is proportional to the logarithm of a; that to a second reading b is proportional to the logarithm of b.

One of the two scales is known as the rule; the other as the slider.

Now let p be the mark on the rule corresponding to a division a, a being the index at the left-hand end of the rule, then AP measures the logarithm of a, so that the number at A is 1. Place c, the index of the slider, which is marked 1, in contact with P, and let Q be the mark on the slider which corresponds to a division b, so that cQ measures log b Let R be the mark on the rule opposite Q, let c be the corresponding reading; then A R= log c. Now

log ab = log a + log b

AP + CQ = AP + PR

=AR = log c;

.. c = ab.

In the figure as drawn, if the distance A B be taken as

unity, then AP is log 3, Q is at division 3 on the slider, and R, the corresponding division on the scale, is 9, which is equal to 3 times 3.

The above result, then, leads to the following method for obtaining the product of two or more quantities by the slide rule :—Thus, if a and b are the quantities, set the index of