an equation which gives the dimensions of force.

Since work is measured by the product of force into the distance through which the force acts, the dimensions of work will be those of force multiplied by length, or

W=MLT-2XL=ML2T-2.

The power (or activity) of an agent is measured by the rate at which it does work; hence the dimensions of power are

ML2T-2

T = ML2T-3.

Knowing the dimensions of these quantities, we can perform the change of units without going through the lengthy reasoning of §§ 7 and 8; we shall indicate the general method to be followed, but it will be best understood by reference to the actual examples given. [See also equations (5) and (6) in § 8.]

Let 9 be any concrete quantity, and let its measure be n in terms of the unit Q, which is based upon the fundamental units M, L, and T; we wish to find its measure n' in a new system in terms of the unit Q' which is based upon M', L', and T'. Since the quantity measured in both systems is the same,

Let the dimensions of Q be M< substituting for O and Q' in equation (7) we have

Ex. 2. Find the number of dynes in a poundal poundal being the British absolute unit of force, b upon the pound, foot, and second).

Referring to equations (7) and (8) we see that, s n = 1, the required number n' is the change-rati multiplier for changing from British to C.G.S. unit force. The dimensions of force are MLT, so x= 1, y = I, and = -2. T and T', the units of ti are the same (one second) in both systems.

Ex. 3. Find the value of a horse-power in watts, a ho power being equivalent to 550 foot-pounds per second, the value of g being 32.18.

As the foot-pound is a gravitation unit, we shall have to reduce to the corresponding absolute unit multiplying by g

550 foot-pounds=550 x 32.18 foot-poundals.

The dimensional equation for finding the equival rate of working in C.G.S. units (ergs per second) is,

550 x 32.18x ML2T-3='x M'L'T'-3.

The units of time (T and T') are the same in b systems; and, as in Example 2,

§ 9

the

sed

nce

or

s of hat me,

10. Magnetic and Electrical Units. The dimensions of the most important of these are given below, and it will be useful practice for the student to deduce them from the corresponding physical laws, as we have done in the preceding articles.

rse

and

first

by

Ex. 4. The dimensions of magnetic intensity (or strength of field) are M1LT, and the horizontal intensity of the earth's magnetic force at Aberystwith is 0.1774 in C.G.S. units: what is its value in British (foot-grain-second) units?

The intensity is the same, whatever units we employ to measure it. Let be its numerical value in the British system, in which the unit of field intensity is H', the corresponding unit in the C.G.S. system being H; then

d, or

and the units of time (T and T') are the same in both

Ex. 5. Assuming Coulomb's law (the law of in squares), to find the dimensions of the unit of quant the electrostatic system.

According to the law of inverse squares, the exerted between two bodies charged with quantit and q' of electricity, and situated at a distance d one another, is proportional to the product of the ch and inversely proportional to the square of the dist Choosing our unit of quantity in accordance with definition of § 4, we may write this in the form

If we suppose that q'q, we have q2= d2 q=df, so that the dimensions of the unit of qua are LXVMLT or MLT

decimals. The degree of approximation to which the calculation must be carried out depends upon the accuracy of the data given. Physical measurements are never absolutely correct. If, then, we have to calculate out the results of an experiment made by a method which is liable to an error of (say) one in a thousand, it would be labour thrown away to carry out the calculation to more than four or five significant figures. Now it frequently happens, in working out the results of physical experiments, that the "uncorrected result" has to be multiplied by one or more correcting factors (each nearly equal to unity) in order to obtain the corrected result ;" and it is to the manipulation of these factors that the student's attention is now directed.

66

Suppose that the experiment under consideration consists in measuring the distance between two points by means of a steel metre scale, the length of which at o° C. is known to be 1.00057 metre; and suppose further that the measurement is carried out at a temperature of 15° C. The steel scale expands on heating, and its length at 15° is greater (Chap. III.) than its length at o° in the ratio of 1.00018 to 1. If the uncorrected distance, as determined by direct measurement, is d, then the true distance (corrected for error of scale and error through temperature) will be

where

d=d(1+a) (1+8),

I + a = 1.00057, and 1+ẞ= 1.00018.

Now (a)(1 + ß) = 1 + a + B+ aß; and since a=0.00057 and B=0.00018, aẞ= 0.000,000, 1026, so that the error caused by neglecting this last term would only be I in 10,000,000. The measurement itself would probably not be correct to 1 in 1,000,000, so that we may safely adopt the approximation, and write

22+a+B) = d x 1.00075.