  (abridged) HCN luminosity Lhcn is a tracer of DENSE molecular gas, n(H_2) >~
3x10^4 cm^{-3}, associated with star-forming giant molecular cloud (GMC) cores.
We present the results and analysis of our survey of HCN emission from 65
infrared galaxies including 9 ultraluminous infrared galaxies (ULIGs,
Lir>10^{12}Lsun), 22 luminous infrared galaxies (LIGs,
10^{11}<Lir<~10^{12}Lsun) and 34 normal spiral galaxies with lower IR
luminosity (most are large spiral galaxies). We have measured the global HCN
line luminosity and the observations are reported in Gao and Solomon (2003,
Paper I). This paper analyzes the relationships between the total far-IR
luminosity a tracer of the star formation rate, the global HCN line luminosity
a measure of the total DENSE molecular gas content, and the CO luminosity a
measure of the total molecular content. We find a tight linear correlation
between the IR and HCN luminosities Lir and Lhcn (in the log-log plot) with a
correlation coefficient R=0.94. The IR--HCN linear correlation is valid over 3
orders of magnitude including ULIGs. The direct consequence of the linear
IR--HCN correlation is that the star formation law in terms of DENSE molecular
gas content has a power law index of 1.0. The global star formation rate is
linearly proportional to the mass of dense molecular gas in normal spiral
galaxies, LIGs, and ULIGs. This is strong evidence in favor of star formation
as the power source in ultraluminous galaxies since the star formation in these
galaxies appears to be normal and expected given their high mass of dense
star-forming molecular gas.
