being the product of moment of inertia by angular velocity, or the product of momentum by length.

Intensity of pressure, or intensity of stress generally,

being force per unit of area, is of dimensions

force

area

; that

Intensity of force of attraction at a point, often called simply force at a point, being force per unit of attracted

equal to the acceleration which it generates, and has accordingly the dimensions of acceleration.

The absolute force of a centre of attraction, better called the strength of a centre, may be defined as the intensity of force at unit distance. If the law of attraction be that of inverse squares, the strength will be the product of the intensity of force at any distance by the square of this distance, and its dimensions will be

1

L3

T2

Curvature (of a curve) = +, being the angle turned by the tangent per unit distance travelled along the curve. Tortuosity, being the angle turned by the osculating plane per unit distance travelled along the curve.

The solid angle or aperture of a conical surface of any form is measured by the area cut off by the cone from a sphere whose centre is at the vertex of the cone, divided by the square of the radius of the sphere. Its dimensions are therefore zero; or a solid angle is a numerical quantity independent of the fundamental units.

The specific curvature of a surface at a given point (Gauss's measure of curvature) is the solid angle described by a line drawn from a fixed point parallel to the normal at a point which travels on the surface round the given point, and close to it, divided by the very small

area thus enclosed. Its dimensions are therefore

1 L2'

The mean curvature of a surface at a given point, in the theory of Capillarity, is the arithmetical mean of the curvatures of any two normal sections normal to each

other. Its dimensions are therefore

1

L

15

CHAPTER II.

CHOICE OF THREE FUNDAMENTAL UNITS.

15. NEARLY all the quantities with which physical science deals can be expressed in terms of three fundamental units; and the quantities commonly selected to serve as the fundamental units are

a definite length,

a definite mass,

a definite interval of time.

This particular selection is a matter of convenience rather than of necessity; for any three independent units are theoretically sufficient. For example, we might em ploy as the fundamental units

a definite mass,

a definite amount of energy,

a definite density.

16. The following are the most important considerations which ought to guide the selection of fundamental units:

(1) They should be quantities admitting of very accurate comparison with other quantities of the same kind.

(2) Such comparison should be possible at all times. Hence the standards must be permanent-that is, not liable to alter their magnitude with lapse of time.

(3) Such comparisons should be possible at all places. Hence the standards must not be of such a nature as to change their magnitude when carried from place to place.

(4) The comparison should be easy and direct.

Besides these experimental requirements, it is also desirable that the fundamental units be so chosen that the definition of the various derived units shall be easy, and their dimensions simple.

17. There is probably no kind of magnitude which so completely fulfils the four conditions above stated as a standard of mass, consisting of a piece of gold, platinum, or some other substance not liable to be affected by atmospheric influences. The comparison of such a standard with other bodies of approximately equal mass is effected by weighing, which is, of all the operations of the laboratory, the most exact. Very accurate copies of the standard can thus be secured; and these can be carried from place to place with little risk of injury.

The third of the requirements above specified forbids the selection of a force as one of the fundamental units. Forces at the same place can be very accurately measured by comparison with weights; but as gravity varies from place to place, the force of gravity upon a piece of metal, or other standard weight, changes its magnitude in travelling from one place to another. A spring-balance, it is true, gives a direct measure of

force;

but its indications are too rough for purposes of

accuracy.

18. Length is an element which can be very accurately measured and copied. But every measuring instrument is liable to change its length with temperature. It is therefore necessary, in defining a length by reference to a concrete material standard, such as a bar of metal, to state the temperature at which the standard is correct. The temperature now usually selected for this purpose is that of a mixture of ice and water (0° C.), observation having shown that the temperature of such a mixture is constant.

The length of the standard should not exceed the length of a convenient measuring-instrument; for, in comparing the standard with a copy, the shifting of the measuring-instrument used in the comparison introduces additional risk of error.

In end-standards, the standard length is that of the bar as a whole, and the ends are touched by the instrument every time that a comparison is made. This process is liable to wear away the ends and make the standard false. In line-standards, the standard length is the distance between two scratches, and the comparison is made by optical means. The scratches are usually at the bottom of holes sunk half-way through the bar.

19. Time is also an element which can be measured with extreme precision. The direct instruments of measurement are clocks and chronometers; but these are checked by astronomical observations, and especially by transits of stars. The time between two successive transits of a star is (very approximately) the time of the

B