Quantity.

DERIVATION OF UNITS ON THE C.G.S. SYSTEM-continued.

Physical law

Variation equation

Definition of derived unit

Derived unit

Dimensional equation

ELECTRO-MAGNETIC UNITS."

The force acting upon a
magnetic pole at the
centre of a circular arc of
wire carrying a current,
is proportional to the
strength of the current,
the length of the wire,
and the strength of the
pole, and inversely pro
portional to the square of
the radius of the arc.
The quantity of electricity
which passes across any
section of a wire is pro-
portional to the current
and to the time of passage.
The work done by a quan-
tity of electricity in passing
between two points is pro-
portional to the quantity
and to the electromotive
force or difference of po-
tential between the points.
The current passing between
two points of a conductor
is directly proportional to
the electromotive force,
and inversely proportional
to the resistance of the
conductor between the
points (Definition of Re-
sistance).
The increase in the potential
of a conductor produced
by the addition of a given
quantity to its charge is in-
versely proportional to the
capacity of the conductor.

1 See chaps. xvii and xviii. The electrostatic system is not employed in the experiments described below.

The C.G.S. System.

The table, p. 18, shows the method of derivation of such absolute units on the C.G.S. system as we shall have occasion to make use of in this book. The first column contains the denominations of the quantities measured; the second contains the verbal expression of the physical law on which the derivation is based, while the third gives the expression of the law as a variation equation; the fourth and fifth columns give the definition of the C.G.S. unit obtained and the name assigned to it respectively, while the last gives the dimensional equation. This will be explained later (p. 24).

The equations given in the third column are reduced to ordinary equalities by the adoption of the unit defined in the next column, or of another unit belonging to an absolute system based on the same principles.

Some physical laws express relations between quantities whose units have already been provided for on the absolute system, and hence we cannot reduce the variation equations to ordinary equalities. This is the case with the formula for the gaseous laws already mentioned (p. 15).

A complete system of units has thus been formed on the C.G.S. absolute system, many of which are now in practical use. Some of the electrical units are, however, proved to be not of a suitable magnitude for the electrical measurements most frequently occurring. For this reason practical units have been adopted which are not identical with the C.G.S. units given in the table (p. 20), but are immediately derived from them by multiplication by some power of 10. The names of the units in use, and the factors of derivation from the corresponding C.G.S. units are given in the following table :—

TABLE OF PRACTICAL UNITS FOR ELECTRICAL MEASUREMENT RELATED TO THE C.G.S. ELECTRO-MAGNETIC SYSTEM.

To shorten the notation when a very small fraction or a very large multiple of a unit occurs, the prefixes micro- and mega- have been introduced to represent respectively division and multiplication by 106. Thus:

Arbitrary Units at present employed.

For many of the quantities referred to in the table (p. 18) no arbitrary unit has ever been used. Velocity, for instance, has always been measured by the space passed over in a unit of time. And for many of them the physical law given in the second column is practically the definition of the quantity; for instance, in the case of resistance, Ohm's law is the only definition that can be given of resistance as a measurable quantity.

For the measurement of some of these quantities, however, arbitrary units have been used, especially for quantities which have long been measured in an ordinary way as volumes, forces, &c.

Arbitrary units are still in use for the measurement of temperature and quantities of heat; also for light intensity, and some other magnitudes.

We have collected in the following table some of the arbitrary units employed, and given the results of experimental determinations of their equivalents in the absolute

units for the measurement of the same quantity when such

Changes from one Absolute System of Units to another.
Dimensional equations.

We have already pointed out that there are more than one absolute system of units in use by physicists. They are deduced in accordance with the same principles, but are based on different values assigned to the fundamental units. It becomes, therefore, of importance to determine the factor by which a quantity measured in terms of a unit belonging to one system must be multiplied, in order to express it in terms of the unit belonging to another system. Since the systems are absolute systems, certain variation equations become actual equalities; and since the two systems adopt the same principles, the corresponding equations will have the constant & equal to unity for each system. Thus, if we take the equation (1) (p. 14) as a type of one of these equations, we have the relation between the numerical measures 9=x" y 27

holding simultaneously for both systems.

Or, if q, x, y, z, be the numerical measures of any quantities on the one absolute system; q', x', y', z', the numerical measures of the same actual quantities on the other system,

Now, following the usual notation, let [o], [x], [v], [z] be the concrete units for the measurement of the quantities on the former, which we will call the old, system, [q'], [x'], [y'], [z] the concrete units for their measurement on the new system.

Then, since we are measuring the same actual quantities,
9 [Q] = q' [Q']
x [x] = x' [x']
y [v] = y' [Y']

. (3).

% [z] = 2' [z']

The symbol is used to denote absolute identity, as distinguished

from numerical equality.