at which energy is given out by a working system; and to avoid circumlocutions in what follows we shall frequently use the term in that sense. Among engineers

in this country the unit rate of working is one horsepower, that is 33,000 foot pounds per minute.

Unit Activity in the C.G.S. system is one erg per second. In practical electricity an activity of 107 ergs per second is frequently employed as unit. This unit has been called a Watt.

Since an Activity is measured by the numerical work done per unit of time, its dimensional formula is given by

[A] = [M L2 T-3].

Energy (E). When a material system in virtue of stresses between its own parts and those of bodies external to it does work or has work done upon it, in passing from one state to another, it is said to give out or to gain energy. The energy given out or gained is measured by the work so done.

If the change be a change of motion, then, according as energy is given out or gained by the system, it is said to lose or gain kinetic energy. If the change be of any other kind, it must be classed under change of configuration, and, according as the system gives out or gains energy, it is usually said to lose or gain potential energy.

When we consider the work done by mutual forces between different parts of the same system, a loss of kinetic energy in the system is accompanied by an equal gain of potential energy, and vice versâ, so that the total energy of the system remains unchanged in amount. principle called the Conservation of Energy.

This is the

Energy is measured by the same units as work, and

its dimensional formula is the same as that of work, that is

[E] = [M L2 T2].

We shall here, for the sake of illustration, give three examples of the application of dimensional formulas to the solution of problems regarding units. The problems are taken from Professor Everett's Units and Physical Constants.

Ex. I. If the unit of time be the second, the unit density 162 lbs. per cubic foot, and the unit of force the weight of an ounce at a place where the change of velocity g produced by gravity in one second is 32 feet per second, what is the unit of length?

Here the change-ratio by which we must multiply the density of a body in the system of units proposed, to find its density in terms of the pound as unit of mass, and the foot as unit of length, is 162. We have therefore, omitting the brackets in the dimensional formulas,

where M is the number of pounds equivalent to the unit of mass, and L the number of feet equivalent to the unit of length. Also, it is plain that the unit of force in the proposed system is two foot-pound-second units. Hence we have also, since T = 1,

By division therefore we get L1=1/81 or L=1/3. The unit of length is therefore 4 inches.

Ex. 2. The number of seconds in the unit of time is equal to the number of feet in the unit of length, the unit of force is 750 lb. weight (g being 32), and a cubic foot of

the substance of unit density contains 13,500 ounces. Find the unit of time.

Using M and L as in the last problem, and putting T for the numerical expression of the unit of time in seconds, we have plainly

That is the unit of time is 5 seconds.

Ex. 3. When an inch is the unit of length and T seconds the unit of time, the numeric of a certain acceleration is a; when 5 feet and I minute are units of length and time respectively, the numeric for the same acceleration is 107.

Find T.

The change-ratio or value of LT-2 for reduction to footsecond units is plainly in the first case T-2/112, in the second 5/3600. We get therefore

1. ELECTROSTATIC SYSTEM.

Quantity of Electricity [9]. In the electrostatic system of units which is convenient when electrostatic results, independently of their bearing on electromagnetic pheno

mena, are required, the units of all the other quantities are founded on the following definition of unit quantity of electricity. Unit quantity of electricity is that quantity which, concentrated at a point at unit distance from an equal and similar quantity, also concentrated at a point, is repelled with unit force when the medium across which the electric action is transmitted is a certain standard insulating medium. An ideal vacuum is sometimes taken as standard, but we shall suppose at present that the medium is air at temp. o° C. and at standard atmospheric pressure. We shall call this simply air.

This definition is precisely similar to the definition (p. 40 above) of unit magnetic pole which forms the basis of another system of units called the electromagnetic system, of much wider and more important application than the electrostatic. Hence by Coulomb's law that (the numerical values of) electric attractions and repulsions are directly as the products of the (numerics for the) attracting and repelling quantities, and inversely as the second power of the (numeric for the) distance between them, if a quantity of positive electricity expressed by q be placed Չ at a point distant L units from an equal quantity of electricity, then the medium being air, the numeric F for the force between them is q2/L2.

If the medium across which the electric action is transmitted be some other medium than air, the force between the charges is numerically q2/KL2 where Kis the numerical measure of a quantity called the electric inductive capacity, or usually the specific inductive capacity of the medium. This quantity is precisely analogous to the conductivity of a substance for heat * and to magnetic

*See Theory and Practice of Absolute Measurements in Electricity and Magnetism, vol. i. chap i. sect. v.

permeability (see p. 350 below). In the ordinary electrostatic system of units this quantity is defined (as at p. 347) so as to have a dimensional formula I, that is to be a mere numeric.

But we might proceed otherwise and regard K as a quantity of undetermined dimensions as regards the fundamental units, but such that q2/KL2 has the dimensions of a force. We may then, in the absence of special reasons for preferring one dimensional formula for K to another, assign its dimensions according to any convenient hypothesis. One such hypothesis is that which forms the basis of the ordinary electrostatic system, namely, that K is, as regards the fundamental units, of zero dimensions, that is has a dimensional formula [1]. But in the ordinary electromagnetic system of units, which has quite a different derivation from the electrostatic, the dimensional formula of K is [L-2T2] and the numerical value of K depends on the choice made of fundamental units.

We shall in what follows suppose the dimensions of K undetermined, and therefore allow the symbol K ́expressing it to appear in the dimensional formulas of the other quantities. We shall thus obtain a more general electrostatic system in which the absolute dimensions of the quantities are not settled. From this the ordinary electrostatic system is obtained by simply deleting K.*

The dimensional formula of quantity of electricity is accordingly [FLK] or [MLT-1K"],

Electric Surface Density [a]. The density of an electric charge on a surface is measured by the quantity of electricity per unit of area. Therefore [o] is [q L-2] or

[M3 L ̃3 T-1 K3].

*This method of proceeding is advocated and its advantages pointed out by Prof. A. W Rücker, F.R.S. in a paper on the "Suppressed Dimensions of Physical Quantities," Phil. Mag., Feb. 1889.