-1 , because the number of units in the velocity of a body is found by dividing the units of length passed over by the units of time occupied in passing. The acceleration of a body is measured by the increase of velocity in relation to the time, that is, we must divide the units of velocity gained by the units of time occupied in gaining it; hence its dimensions will be LT -2 . Momentum is the product of mass and velocity, so that its dimensions are MLT -1 . The effect of a force is measured by the acceleration produced in a unit of mass in a unit of time; hence the dimensions of force are MLT -2 . Work done is proportional to the force acting and to the space through which it acts; so that it has the dimensions of force with that of length added, giving ML 2 T -2 .
It should be particularly noticed that angular magnitude has no dimensions at all, being measured by the ratio of the arc to the radius (p. 305). Thus we have the dimensions LL-1 or L0. This agrees with the statement previously made, that no arbitrary unit of angular magnitude is needed. Similarly, all pure numbers expressing ratios only, such as sines and other trigonometrical functions, logarithms, exponents, &c., are devoid of dimensions. They are absolute numbers necessarily expressed in terms of unity itself, and are quite unaffected by the selection of the arbitrary physical units. Angular magnitude, however, enters into other quantities, such as angular velocity, which has the dimensions 1/T or T-1, the units of angle being divided by the units of time occupied. The dimensions of angular acceleration are denoted by T-2.
The quantities treated in the theories of heat and electricity are numerous and complicated as regards their dimensions. Thermal capacity has the dimensions ML-3, thermal conductivity, ML-1T-1. In Magnetism the dimensions of the strength of pole are M1/2L3/2T-1, the dimensions of field-intensity are M1/2L-1/2T-1, and the intensity of magnetisation has the same dimensions. In the science of electricity physicists have to deal with numerous kinds of quantity, and their dimensions are different too in the electro-static and the electro-magnetic systems. Thus electro-motive force has the dimensions M1/2L1/2T-1, in the former, and M1/2L3/2T-2 in the latter system. Capacity simply depends upon length in electro-statics, but upon L-1T2 in electro-magnetics. It is worthy of particular notice that electrical quantities have simple dimensions when expressed in terms of density instead of mass. The instances now given are sufficient to show the difficulty of conceiving and following out the relations of the quantities treated in physical science without a systematic method of calculating and exhibiting their dimensions. It is only in quite recent years that clear ideas about these quantities have been attained. Half a century ago probably no one but Fourier could have explained what he meant by temperature or capacity for heat. The notion of measuring electricity had hardly been entertained.
Besides affording us a clear view of the complex relations of physical quantities, this theory is specially useful in two ways. Firstly, it affords a test of the correctness of mathematical reasoning. According to the Principle of Homogeneity, all the quantities added together, and equated in any equation, must have the same dimensions. Hence if, on estimating the dimensions of the terms in any equation, they be not homogeneous, some blunder must have been committed. It is impossible to add a force to a velocity, or a mass to a momentum. Even if the numerical values of the two members of a non-homogeneous equation were equal, this would be accidental, and any alteration in the physical units would produce inequality and disclose the falsity of the law expressed in the equation.
Secondly, the theory of units enables us readily and infallibly to deduce the change in the numerical expression of any physical quantity, produced by a change in the fundamental units. It is of course obvious that in order to represent the same absolute quantity, a number must vary inversely as the magnitude of the