350 BUCKINGHAM: PHYSICALLY SIMILAR SYSTEMS 



independent and not derivable from one another, is reducible to 

 the form 



^ (tti, t2, ■ • ■ Tn-k, r', r", ■ ■ etc.) = (8) 



in which the r 's are all the independent ratios of quantities of the 

 same kind, and each ir is determinable from an equation of the 

 form 



W] = [Q^Q'2 ■ ■ • -QlPJ-i-^] (9) 



The independent arguments of ^ in equation (8), including 

 both the tt's and the r's, are all the independent dimensionless 

 products of powers of all the quantities involved in the relation, 

 whether of different kinds or not. If the whole number of quanti- 

 ties is h, the number of these independent arguments is (h — k). 

 Hence any complete physical equation is reducible to the form 



^ (Zi, X„ X,_,) (10) 



in which the X's are all the possible independent dimensionless 

 combinations of powers of all the quantities involved. 



5. In practice, the r 's are evident upon inspection of the whole 

 list of quantities; there is therefore no occasion for finding them 

 from equations of the form (9) , and equation (8) is more conven- 

 ient than the symmetrical form (10), which does not distinguish 

 between the r's, which are pure ratios with absolute numerical 

 values, and the expressions r which, while dimensionless and inde- 

 pendent of the sizes of the fundamental units of our system, do 

 depend on the definitions according to which the derived and 

 fundamental units of the absolute system are connected. 



It often happens that one or more of the tt's are also evident 

 upon inspection. In this event, only the remaining tt's need be 

 found by the routine process of solving equation (9) for the ex- 

 ponents. 



When the solution of equation (9) results in a value of t which 

 is inconvenient to write, we are at liberty to replace the expression 

 found, by any function of itself: for this new expression will still 

 be dimensionless and independent of the other t 's. This remark 

 enables us to dispense with the fractional exponents which some- 

 times result from the solution of equation (9) . 



