Principles of the Theory of Similarity

Abstract

Since the numerical values of the dimensionless quantities remain the same in all systems of fundamental units, the numerical values of the dimensionless quantities X _j and Π _A , encountered in the preceding chapter, will not change if the system of fundamental units L″, T″ and M″ by which they are expressed is replaced by a different system of units L″, T″ and M″. Two systems of fundamental units can always be related to each other by the ‘transformation formulae’

$$ \begin{array}{*{20}{c}} {L'' = {\lambda _L}L'} \\ {T'' = {\lambda _T}T'} \\ {M'' = {\lambda _M}M'} \end{array} $$

((2.1))

where the proportionality factors λ _L , λ _T and λ _M are some arbitrary constants. Accordingly, the dimensionless numbers in general (and X _j and Π _A in particular) can be regarded as the quantities which remain invariant with respect to the transformation (2.1).