BUCKINGHAM: PHYSICALLY SIMILAR SYSTEMS 349 



(n—k) being derived from them. Let [Qi],.---[Q/k] be such a set, 

 and let [QA.+i]v-[Qn] be denoted by [Pi],....[P„_J. Each unit 

 [P] is connected with or derivable from the [Q] 's by a dimensional 

 equation which may be written in the form 



[Qrol ■■■■Qip] = m (6) 



Since there are (n — k) of the P's, there are (n — k) independent 

 equations of this sort, and no more. 



If in equation (6) we substitute for [P] and the [Q]'s their 

 dimensional equivalents in terms of any convenient fundamental 

 units — which will necessarily be k in number — the requirement 

 that the exponent of each fundamental unit shall vanish separately 

 furnishes k independent linear equations which suffice for the 

 determination of the exponents a, /S, etc. If, after determining 

 these exponents for any particular [P], we set 



- = QrQf • • • q:p (7) 



T satisfies the condition of being a dimensionless product of the 

 required form. There are (n — k) independent equations of the 

 form (6), one for each of the quantities P, and the same number of 

 independent tt's; hence i = n — k. 



4. We have hitherto confined our attention to a relation among 

 quantities that are all of different kinds. If several quantities of 

 any one kind are involved in the relation to be described, they may 

 all be specified by the value of any one and the ratios r', r" , etc., 

 of the others to that one. Dimensional considerations cannot 

 tell us anything about the manner in which these dimensionless 

 ratios r appear in the equation which describes the relation; but 

 their possible influence must be indicated, and this may be done 

 in an entirely general way by introducing them as additional 

 independent arguments of the unknown function ^. The limita- 

 tion imposed by the requirement of dimensional homogeneity 

 upon the possible forms of physical equations may therefore be 

 conveniently summarized in the following statement : 



Any complete physical equation which describes a relation sub- 

 sisting among quantities of n different kinds, of which k kinds are 



