352 BUCKINGHAM: PHYSICALLY SIMILAR SYSTEMS 



The chief value of the principle of dimensional homogeneity- 

 is found in its application to problems in which it is possible to 

 arrange matters so that the r's and tt's of equation (11) shall 

 remain constant and the definite equation (13) therefore be 

 satisfied. 



8. The quantities involved in a physical relation pertain to 

 some particular physical system which may usually be treated as 

 of very limited extent. Let S be such a system, and (11) the 

 equation which describes a relation subsisting among certain 

 quantities of the kinds Q and P which pertain to S, e.g. the sizes, 

 densities, thermal conductivities, etc. of its essential parts. 



Let S' be a second system into which S would be transformed 

 if all quantities of each kind Q involved in equation (11), were 

 changed in some arbitrary ratio, so that the r 's for -all quantities 

 of these kinds remained constant, while the particular quantities 

 Qi, Q2,-"'Qk changed in k independent ratios. For example; if 

 Qi is a length, S and S' are to be geometrically similar in all their 

 essential parts, though other parts, of which the size and shape 

 have no influence on the relation under consideration, are not 

 subject to any geometrical conditions. The systems now ''corre- 

 spond" as regards their essential parts and may be said to be 

 similar as regards each of the kinds of quantity Q separately, so 

 far as such quantities are involved in equation (11). 



In addition to the foregoing specifications, which relate to the 

 changes of the Q's during the transformation of S into S', let P2 

 and all quantities of that kind involved in the relation, change in 

 such a ratio, dependent on the arbitrary changes of the Q's, that 

 7r2 remains constant; and let similar conditions be imposed on 

 P3, Pi,....Pn-k, and all quantities of those kinds involved in the 

 relation. Two systems S and S' which are related in the manner 

 just described are similar as regards the physical relation in ques- 

 tion. 



If two systems correspond in the manner described for the k 

 independent kinds of quantity Q, and if it is practicable to make 

 them similar by fulfilling the conditions which relate to the 

 quantities of the kinds P, equation (13) is satisfied by the quanti- 

 ties pertaining to either system, and the value of N determined by 

 experiment on one system may at once be used for the other. 



