A star undergoes some mode of oscillation. How does the frequency 
of oscillation depend upon the properties of the star? The first step is the
identification of the physically relevant variables. Certainly the density 
and the radius R are important; we'll also need the gravitational
constant G which appears in Newton's law of universal gravitation. We
could add the mass m to the list, but if we assume that the density is
constant as a first approximation, then 
, and the mass
is redundant. Therefore, is the governed parameter, with dimensions
, and 
are the governing parameters, with
dimensions 
, [R]=L, and 
(check the last one). You can easily check that have
independent dimensions; therefore, n=3, k=3, so the function 
is
simply a constant in this case. Next, determine the exponents:
Equating exponents on both sides, we have
Solving, we find a=c=1/2, b=0, so that
with C a constant. We see that the frequency of oscillation is proportional to the square root of the density, and independent of the radius. Once again, the determination of C requires a real theory of stellar oscillation, but the interesting dependence upon the physical parameters has been obtained from dimensional considerations alone.
