348 BUCKINGHAM: PHYSICALLY SIMILAR SYSTEMS 



in which the coefficients M are dimensionless or pure numbers. 

 No purely arithmetical operator, such as log or sin can be applied 

 to an operand which is not a pure number; and whenever functions 

 that are not expressible as sums of terms of the form (1) occur in 

 physical equations, their arguments are always dimensionless 

 numbers. The results of the indicated operations are therefore 

 also dimensionless numbers, and such functions, when they 

 appear, may be included in the dimensionless coefficients M. 



2. Upon dividing equation (1) through by any term we have 



i + X^QTQT • • • q:« = (2) 



All the terms of a physical equation must have the same dimen- 

 sions, and the A'''s are of zero dimensions; hence the exponents a, 

 of each term, must be such that a dimensional equation 



[qT QT • ■ ■ QT] = [1] (3) 



is satisfied. 



Let TT represent any dimensionless product of the form defined 



by equation (3). Then tt"^ is also dimensionless; and if tti, 7r2, 



TT,- are all the independent dimensionless products which can be 

 made by combining powers of the Q's, equation (2) may still 

 satisfy the dimensional requirement by having the more general 

 form 



1+ ^NtI'tI' • • • • T^' =0 (4) 



Since the number of terms and the values of the A^'s and x's 

 are indeterminate, the S is merely some entirely unknown function 



of the independent arguments iri, tto, ir^. Hence the most 



general form that equation (1) can have, subject to the dimen- 

 sional conditions, is 



^/ (tTi, tTs, • • • • TTi) = (5) 



3. Let k be the number of fundamental units needed in an abso- 

 lute system for measuring the n kinds of quantity: then among 

 the n units required, there is always at least one set of k which 

 are independent and not derivable from one another, and which 

 might therefore be used as fundamental units, the remaining 



