On Physically Similar Systems; Illustrations of the Use of Dimensional Equations

Abstract

ired to descri be by an «i;quation, a relation which subsists among a number of ph ysical quantit i es . of n different kinds.If several quantities of any one kind are involved in the relation, let them be sp ecified by the value of any one and the ratios of the others to this one.The equation w ill then c onta i n n symbols Qi • • • Qn, on e for each kind of quantity, and also, in general, a number of ratios r ' , r " , etc., so that it may he wr itte n(1)Let us suppose, for the present only, that the ratios r do not vary d u ring the ph e nomenon described by the equation : for example, if the equat io n describes a property of a material system and innJln:s lengths, th e system shall remain geometrically s i mi la r to itself during any changes of size wh ich may occur.Vnder this condition equation (1) reduces to(2)If none of th e qua nt it ies involved in the relation has been overlooked, the e quation will gi ve a co mplete desc r i pt io n of the re la t i on subsisting among the quant i ties represe nte d in it, and will be a complete equation.Th e co efficie nt s of a complete equation are dimensionless numbers, i. e., if the quant i t i es Q are m e asured by an absolute system of units, the coef fici e nts of the e quation do not depend on the sizes of the fundamental units but only on th e fixe d in t er re l ati on s of the units which characterize the system and differentiat e it from any other absolute system.To illustrate w hat is m e ant by a " com ple te "equation, we may consider the fam i liar e quat io n � = constant, in which p is the pressure, fl the specific volume, and 8 the absolute temper ature of a mass of gas.The constant is not dimensionless but depends, even for a g i ven ga s, on th e units adopted for measuring p, v, and 8; the equation is not compl ete .Further investigation shows that VoL.IV.] No.4.