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 DIMENSIONS all departments of physical science that are reducible to pure dynamics. The mode of transformation of a derived entity, as regards its numerical value, from one set of fundamental units of reference to another set, is exhibited, in the simple illustrations above given. When the numerical values of the new units, expressed in terms of the former ones, are substituted for the symbols, in the expression for the dimensions of the entity under consideration, the number which results is the numerical value of the new unit of that entity in terms of the former unit: thus all numerical values of entities of this kind must be divided by this number, in order to transfer them from the former to the latter system of fundamental units. As above stated, physical science reduces the phenomena of which it treats to the common denomination of the positions and movements of masses. Before the time of Gauss it was customary to use a statical measure of force, alongside the kinetic measure depending on the acceleration of motion that the force can produce in a given, mass. Such a statical measure could be conveniently applied by the extension of a spring, which, however, has to be corrected for temperature, or by weighing against standard weights, which has to be corrected for locality. On the other hand, the kinetic measure is independent of local conditions, if only we have absolute scales of length and time at our disposal. It has been found to be indispensable, for simplicity and precision in physical science, to express the measure of force in one way 5 and statical forces. are therefore now generally referred in theoretical discussions to the kinetic unit of measurement. In mechanical engineering the static unit has largely survived; but the increasing importance of electrical applications is introducing uniformity there also. In the science of. electricity two different systems of units, the electrostatic and the electrodynamic, still to a large extent persist. The electrostatic system arose because in the development of the subject statics came before kinetics; but in the complete synthesis it is even now found convenient to express the various quantities in terms of the electrokinetic system alone. The system of measurement now adopted as fundamental in physics takes the centimetre as unit of length,. the gramme as unit of mass, and the second as unit of time. The choice of these units was in the first instance arbitrary and dictated by convenience; for some purposes subsidiary systems based on multiples of these units by certain powers of ten are found convenient. There are certain absolute entities in nature, such as the constant of gravitation, the velocity of light in free space, and the constants occurring in the expression giving the constitution of the radiation in an enclosure that corresponds to each temperature, which are the same for all kinds of matter; these might be utilized, if known with sufficient accuracy, to establish a system of units of an absolute. or cosmical kind. The wave-length of a given spectral line might be utilized in the same manner, but that depends on recovering the kind of matter which produces the line. In physical science the uniformities in the course of phenomena are elucidated by the discovery of permanent or intrinsic relations between the measurable properties of material systems. Each such relation is expressible as an equation connecting the numerical values of entities belonging to the system. Such an equation, representing as it does a relation between actual things, must remain true when the measurements are referred to a new set. of fundamental units. Thus, for example, the kinematical equation v’ = nf H, if n is purely numerical, contradicts the necessary relations involved in the definitions of the entities velocity, acceleration, and length which occur in

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it. For on changing to a new set of units as above the equation should still hold; it, however, then becomes /ryB = n. l'FY • £/[L]. Hence on division there remains a dimensional relation [V]2 = [F]2[L], which is in disagreement with the dimensions above determined of the derived units that are involved in it. The inference follows either that an equation such as that from which we started is a formal impossibility, or else that the factor n which it contains is not a mere number, but represents n times the unit of some derived quantity which ought to be specified in order to render the equation a complete statement of a physical relation. On the latter hypothesis the dimensions [N] of this quantity are determined by the dimensional equation J_V]2 = [LJ where in terms of the fundamental units of length and time, rVl = [L][T]-1, [F] = [L][T]"2; whence by substitution it appears that [N] = [L]_1[T]2. Thus, instead of being merely numerical, n must represent in the above formula the measure of some physical, entity, which may be classified by the statement that it has the conjoint dimensions of time directly and of velocity inversely.. It often happens that simple comparison of the dimensions of the quantities which determine a physical system will lead to important knowledge as to the necessary relations that subsist between them. Thus in the case of a simple pendulum the period of oscillation r can depend only on the angular amplitude a of the swing, the mass m of the bob considered as a point, and the length l of the suspending fibre considered as without mass, and on the value of g the acceleration due to gravity, which is the active force; that is, r —fig, h fj)- The dimensions must be the same on both sides of this formula, for, when they are expressed in terms of the three independent dynamical quantities mass, length, and time, there must be complete identity between its two sides. Now the dimensions of g are [L][T]'2; and when the unit of length is altered, the numerical value of the period is unaltered, hence its expression must be restricted to the form /(a, m, l/g). Moreover, as the period does not depend on the unit of mass, the form is further reduced to /(a, l/g); and as it is of the dimensions +1 in time, it must be a multiple of (l/g)*, and therefore of the form cf>(a) s](ljg). Thus the period of oscillation has been determined by these considerations except as regards the manner in which it depends on the amplitude a of the swing. When a process of this kind leads to a definite result, it will be one which makes the unknown quantity jointly proportional to various powers of the other quantities involved; it will therefore shorten the process if we assume such an expression for it in advance, and .find whether it is possible to determine the exponents definitely and uniquely so as to obtain the correct dimensions. In the present example, assuming in this way the relation T = kapmqlrgs, where A is a pure numeric, we are led to the dimensional equation [T] = [aj^Mj^Lj^LT'2]8, showing that the law assumed would not persist when the fundamental units of length, mass, and time are altered, unless y = 0, s = - |, r = ^ ; as an angle has no dimensions, being determined by its numerical ratio to the invariable angle forming four right angles, p remains undetermined. This leads to the same result, T = ^(a)^+*y-, as before. As illustrating the power and also the limitations of this method of dimensions, we may apply it (after Lord Rayleigh, Aoy. Soc. Proc., March 1900) to the laws of viscosity in gases. I lie dimensions of1 viscosity (/x) are (force/area)-r(velocity/lengt), giving IML-W- ! in terms of the fundamental units._ Now, on the dynamical theory of gases viscosity must be a function of the mass mol a molecule, the number n of molecules per unit volume, then velocity of mean square v, and their effective _ radms a it can depend on nothing else. The equation of dimensions cannot supply more than three relations connecting these four possibilities of