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(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 65infrared 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 IRluminosity (most are large spiral galaxies). We have measured the global HCNline luminosity and the observations are reported in Gao and Solomon (2003,Paper I). This paper analyzes the relationships between the total far-IRluminosity a tracer of the star formation rate, the global HCN line luminositya measure of the total DENSE molecular gas content, and the CO luminosity ameasure of the total molecular content. We find a tight linear correlationbetween the IR and HCN luminosities Lir and Lhcn (in the log-log plot) with acorrelation coefficient R=0.94. The IR--HCN linear correlation is valid over 3orders of magnitude including ULIGs. The direct consequence of the linearIR--HCN correlation is that the star formation law in terms of DENSE moleculargas content has a power law index of 1.0. The global star formation rate islinearly proportional to the mass of dense molecular gas in normal spiralgalaxies, LIGs, and ULIGs. This is strong evidence in favor of star formationas the power source in ultraluminous galaxies since the star formation in thesegalaxies appears to be normal and expected given their high mass of densestar-forming molecular gas.Comment: ApJ, May 1 issue (final version, 20 pages + 9 figures + 3 tables +  appendix/refs.). Please see Gao & Solomon (Paper I, astro-ph/0310341) for the  presentation and discussion of the HCN surve

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