The Rayleigh's method of dimensional analysis becomes more laborious if the variable are more than the number of fundamental dimensions (M,L,T).

This difficulty is over come by Buckingham's $\pi$-theorem which states , "If there are n variables in a physical phenomenon and if these variables contain m fundamentals dimensions (M,L,T) then the variables are arranged into (n-m) dimensionless terms Each terms is called $\pi$ -term".

Let $x_{1},x_{2} \ and \ x_{3}...x_{n}$ be the variables involved in a physical problem. Let $x_{1}$be the dependent variable and $x_{2},x_{3}...x_{n}$ are the independent variables on which $x_{1}$ depends and then $x_{1}$ is a function of $x_{2},x_{3}....x_{n}$ and mathematically it is expressed as

$x_{1}=f(x_{2},x_{3}...x_{n})$----- (1)

It can also be written as

$F_{1}(x_{1},x_{2},x_{3}....x_{n})$=0---- (2)

Eqn no 2 is a dimensionally homogeneous equation. It contains 'n' variables. If there are m fundamentals dimensions then according to Buckingham's $\pi$ theorem

$f(\pi_{1},\pi_{2}.....\pi_{n-m})$=0

Each $\pi$ term is dimensionless and is independent of the system. Each $\pi$ term contains m+1 variable.

Let in the above case $x_{2},x_{3} \ and \ x_{4}$ are repeating variables if the fundamental dimension m(M,L,T)=3 then each $\pi$ term is written as

$\pi_{1}=x_{2}^{a_{1}}, x_{3}^{b_{1}}.x_{4}^{c_{1}}.x_{1}$

$\pi_{2}=x_{2}^{a_{2}}, x_{3}^{b_{2}}.x_{4}^{c_{2}}.x_{5}$

$\pi_{n-m}=x_2^{a_n-m} x_3^{b_n-m} x_4^{c_n-m} X_n$

Each equation is solved by homogeneity equation and values of a,b,c etc are obtained. the obtained values are substituted in above equation. and values of $\pi_{1},\pi_{2},\pi_{3}....\pi_{n-m}$ are obtained

These values of $\pi$ are then put in equation no (2) $\pi_{1}=\Phi[\pi_{2},\pi_{3}....\pi_{n-m}]$

$\pi_{2}=\Phi_{1}[\pi_{1},\pi_{3}....\pi_{n-m}]$