DIMENSIONS. All physical quantities. such as force, energy. electric intensity. magnetic poles, etc., admit of mathematical expression in terms of the elementary ideas of physics. Thus, all mechanical quantities can be expressed in terms of mass. length, and duration of time. All electrical and imometic quantities can he ex pressed in of mass. length, duration of time, and the Induetivity' of matter for electric nr maLmetie forces. (See Ett-a-rnicfrv.) Thus, velocity is defined as the limiting value of Ar to A t where A* is the distance traversed in the time Pt; consequently velocity involves the idea of a number of units of length divided by a number of time-units: its 'dimensions' are said to be or LT . If, therefore, on any system of units, e.g. the C.G.S. system (q.v.), the numerical value of a certain velocity is V, its value on a system of units in which the unit of length is ten times as largo as in the former system would be 1710. Similarly, acceleration or the rate of change of velocity with reference to the time has the dimensions (1-, ) T or LT'. Force is measured by the product of acceleration and mass; its dimensions are there fore In the following table are given the dimen sions of various mechanical quantities: acceleration Force. Al L Pressure L Work 1 Al Energy of force AI If K and A are written for the dimensions of electric and magnetic inductivity. the dimen sions of electric and magnetic quantities have the following values. In the first column are given the eleetro-static system of units; in the second. the eleetro-magnetic. (See ELECTRICITY.) Electric quantity AI I T -1 K / or L p./ Electric current T K / or L AI _Magnetic pole I 3 Al T a / Electric resistance L T Electromotive force Li Al/ or Al T Electric capacity. LK
The dimensions of any one physical quantity, however expressed, must be identical; and there fore. choosing electric current L MI = Al/ or Hence L This means that, although the dimensions of neither electrical nor magtietie induetivity are known in terms of length, mass. or time, the product of the two has the dimensions T 4; i.e. the same dimensions as the square root of the reciprocal of a velocity.
It should be noted that work and moment of force have the same dimensions; but there is a difference in this respect. in work the element of distance is in the direction of the force, while in moment of force the element of distance is at right angles to the direction of the force. This might be indicated by calling X and Y the di mensions of length in directions at right angles to each other; in which case force has the dimen sions work, moment of force, A1XYT in any equation connecting physical quanti ties, it is evident that the dimensions of the quantities on the two side: of the equation must he identical; for a mass cannot equal a length, etc. This fact is often useful either in verifying general conclusions or in predicating the con nection between various quantities. The subject of dimensions is thoroughly treated in Daniel's Text-Book of the Principles of Physics (London, 1894) ; Maxwell, //eat (London, 1891) ; Everett, The C. G. S. System (London, 1875).