Turn again to the theoretic side. To take C as proportional to k does away temporarily with the necessity of answering the question as to the dimensions of temperature. Nevertheless the question calls for an answer. If temperature can be identified with the energy of agitation its specification involves a velocity. If it be so expressed in the equation for the transfer of heat a different set of dimensions is obtained: this uncertainty as to dimensions throws doubt on the method. This point has been discussed by Lord Rayleigh and others (see Nature, 1915) but without it being cleared up satisfactorily. The real cause of the difference seems to be that when temperature is recognized as involving molecular velocity there are two velocities entering into the problem—the velocity of molecular agitation and the velocity of the stream of fluid. In order to obtain true dynamical similarity both these velocities must be varied in the same propor tion; in other words the temperature must be varied as the square of the stream velocity. The solution given above corresponds to the case of constant temperature: that is the stream velocity changes without any change in the velocity of agitation. To pass from one case to another is like passing from similar tuning forks to those that are not similar (see above).
But electricity can be measured in other ways. Magnetic forces between poles are measured in units depending on an inverse square law of force, F = MM'/ where Aagain depends on the medium. If we take /.4 as having no dimensions the dimensions of M are In addition electric currents are measured in terms of the force on unit pole when flowing in a circuit of unit radius; and a current has the dimensions of an electric charge divided by a time. It is clear that from such data a
second unit of charge is definable and its dimensions are Ali Li; it is called the electromagnetic unit of charge. In this case indeterminateness arises since p. for vacuum may be a mere number or may represent another property of vacuum. What we measure as a pole of strength M may perhaps be definable as mo/mo where is a more fundamental physical quantity inde pendent of the medium.
It follows from this summary that it is not merely a question of choosing two units of different sizes (like an inch and a metre for lengths) but of choosing two units which fail to satisfy the law of dimensional homogeneity. Homogeneity can be brought about by regarding and ,uo (both for vacua) as not being mere numbers but as having such dimensions that µ Li is homogeneous with This requires that = a numeric or in other words, that AK has the dimensions of the reciprocal of the square of a velocity. This velocity has been shown to be that with which electric waves are propagated in the medium (see ELECTRIC WAVES) ; its value for vacuum is equal to the ratio of an electromagnetic unit of charge to an electrostatic unit, that is (as nearly as is known) 3 X 'oil' cm./sec. In this article we are not concerned with the details of electric measurements but with applications of the law of dimensional homogeneity. On the two systems of units we treat either K or else A as a numeric. Their real dimensions we do not know; those of their product alone are known. We know neither the proper ties of the medium which the symbols represent nor (in conse quence) the physical nature of an electric charge or of a magnetic pole. The situation from the point of view of dimensions can be best illustrated by an example.
Suppose that in the flow of liquids through tubes we had found that its character depended upon the liquid not owing to differ ence of density alone but that another factor was necessary which varied from liquid to liquid, e.g., the viscosity which we assume we have not learned how to express in terms of mechanical data and which we treat as a numeric. Dimensionally we write Whenµ is constant the critical velocity is therefore given either by or = const. The latter is the result arrived at by ignoring the dimensions of The general theorem just given enables one, however, to go further and discuss the cases where p, is a variable.
In very much the same way, limited applications of the law of homogeneity can be made in electromagnetic problems. For example, the time required for a current to fall to of of its value in a linear, conducting, electric circuit is directly as the self-inductance and inversely as the resistance, both being measured in electromagnetic units (or both in electrostatic units).