Let the relevant physical quantieis - surface tension, viscosity, thermal conductivity etc - be
\[P_1, \: P_2, \: ... , \: P_n\]
and let the fundamental quantities - time, length, temperaure etc - be \[m_1, \: m_2, \: ... , \: m_k\]
.We can write
\[P_j=m_1^{a_{1j}}m_2^{a_{2j}}...m_k^{a_{kj}} , \: j=1, \:2, \: ..., \: n\]
.We can write a dimensional matrix
\[A= \{a_{ij} \}\]
.Considered as a vector space, if the number of linearly independent vecors is
\[r\]
, each of the remaining \[n-r\]
vectors can be expressed as a linear combination of the \[r\]
independent vectors, so \[ P_j = \sum_{i=1}^r w_{ij} P_i , \: j=r+1, \: r+2, \: n\]
where the \[w_{ij}\]
are constants.In terms of the physical quantities this becomes
\[P_j = \prod_{i=1}^r P_i^{w_{ij}}\]
.
There will be \[n-r\]
dimensionless groups. This means that any function of all the \[m_i\]
can be expressed in terms of \[n-r\]
dimensionless \[P_j\]
.