relation between the numerical measures of the quantities involved, and hence if one of the units of measurement is changed, the numerical measure of the same actual quantity will be changed in the inverse ratio, and the value of k will be thereby changed.

in

We may always determine the numerical value of k if we can substitute actual numbers for q, x, y, z, the equation (1).

...

For example, the gaseous laws may be expressed in words thus:

'The pressure of a given mass of gas is directly proportional to the temperature measured from -273° C., and inversely proportional to the volume,' or as a variation equation

We may determine k for 1 gramme of a given gas, say hydrogen, from the consideration that 1 gramme of hydrogen, at a pressure of 760 mm. of mercury and at o° C., occupies 11200 cc.

Here has been expressed in terms of the length of an equivalent column of mercury; and thus, if for v and we substitute in equation (2) the numerical measures of any volume and temperature respectively, we shall obtain the corresponding pressure of 1 gramme of hydrogen expressed in millimetres of mercury.

This, however, is not the standard method of expressing

a pressure; its standard expression is the force per unit of area. If we adopt the standard method we must substitute for p not 760, but 76 × 136 × 981, this being the number of units of force' in the weight of the above column of mercury of one square-centimetre section. We should then get for k a different value, viz.:—

and now substituting any values for the temperature and volume, we have the corresponding pressure of 1 gramme of hydrogen expressed in units of force per square centimetre.

Thus, in the general equation (1), the numerical value of k depends upon the units in which the related quantities are measured; or, in other words, we may assign any value we please to k by properly selecting the units in which the related quantities are measured.

It should be noticed that in the equation.

q=kx" y z ...

.

we only require to be able to select one of the units in order to make k what we please; thus x, y, z, . . may be beyond our control, yet if we may give q any numerical value we wish, by selecting its unit, then k may be made to assume any value required. It need hardly be mentioned that it would be a very great convenience if k were made equal to unity. This can be done if we choose the proper unit in which to measure Q. Now, it very frequently happens that there is no other countervailing reason for selecting a different unit in which to measure Q, and our power of arbitrary selection of a unit for Q is thus exercised, not by selecting a particular quantity of the same kind as Q as unit, The units of force here used are dynes or C.G. S. units of force.

and holding to it however other quantities may be measured, but by agreeing that the choice of a unit for Q shall be determined by the previous selections of units for X, Y, Z, . . . together with the consideration that the quantity k shall be equal to unity.

Fundamental Units and Derived Units.

It is found that this principle, when fully carried out, leaves us free to choose arbitrarily three units, which are therefore called fundamental units, and that most of the other units employed in physical measurement can be defined with reference to the fundamental units by the consideration that the factor k in the equations connecting them shall be equal to unity. Units obtained in this way are called derived units, and all the derived units belong to an absolute system based on the three fundamental units.

Absolute Systems of Units.

Any three units (of which no one is derivable from the other two) may be selected as fundamental units. In those systems, however, at present in use, the units of length, mass, and time have been set aside as arbitrary fundamental units, and the various systems of absolute units differ only in regard to the particular units selected for the measurement of length, mass, and time. In the absolute system adopted by the British Association, the fundamental units selected are the centimetre, the gramme, and the second respectively, and the system is, for this reason, known as the C.G.S. system.

For magnetic surveying the British Government uses an absolute system based on the foot, grain, and second; and scientific men on the Continent frequently use a system based on the millimetre, milligramme, and second, as fundamental units. An attempt was also made, with partial success, to introduce into England a system of absolute units, based upon the foot, pound, and second as fundamental units.

C

Variation
equation

Definition of derived unit

Derived unit

Dimensional equation

FUNDAMENTAL ARBITRARY, UNITS

The areas of similar figures
are proportional to the
second power of their
linear dimensions.

The volumes of similar solids
are proportional to the
third power of their linear
dimensions.

The mass of a body is pro-
portional to its volume
and its density conjointly
(Definition of Density).
The space passed over by a
body moving uniformly is
proportional to thevelocity
and the time of passage.
The increase in the velocity
of a body moving with a
constant acceleration is
proportional to the ac-
celeration and the time
during which the motion
has been accelerated.
The force which produces
change of motion in a
body is proportional to
the rate of change of mo-
mentum produced (the rate
of change of momentum
being measured numeri-
cally by the product of the
numerical measure of the
mass of the body, and the
numerical measure of the
acceleration produced).
(Second Law of Motion.)

ναρ

Centimetre. cm.

Gramme. gm.

Second. sec.

The area of a square the length of a side of which is i centimetre.

Sq. cm.

The volume of a cube the C.c. length of a side of which

is I centimetre.

Volume elas. The fractional diminution -2 _ þa− k、

in volume of a fluid under
increased pressure is pro-
portional directly to the
increase of pressure and
inversely to the elasticity
of the fluid.

The moment of a solenoidal
magnet with poles of 1
C.G S. unit strength and
length 1 centimetre.