Polar substituent constants describe the way a substituent will influence a reaction through polar (inductive, field, and resonance) effects.  To determine σ Taft studied the hydrolysis of methyl esters (RCOOMe).  The use of ester hydrolysis rates to study polar effects was first suggested by Ingold in 1930.  The hydrolysis of esters can occur through either acid and base catalyzed mechanisms, both of which proceed through a tetrahedral intermediate.  In the base catalyzed mechanism the reactant goes from a neutral species to negatively charged intermediate in the rate determining (slow) step, while in the acid catalyzed mechanism a positively charged reactant goes to a positively charged intermediate.
Due to the similar tetrahedral intermediates, Taft proposed that under identical conditions any steric factors should be nearly the same for the two mechanisms and therefore would not influence the ratio of the rates.  However, because of the difference in charge buildup in the rate determining steps it was proposed that polar effects would only influence the reaction rate of the base catalyzed reaction since a new charge was formed.  He defined the polar substituent constant σ* as:
where log(k/k is the ratio of the rate of the base catalyzed reaction compared to the reference reaction, log(k/k is ratio of a rate of the acid catalyzed reaction compared to the reference reaction, and ρ* is a reaction constant that describes the sensitivity of the reaction series.  For the definition reaction series, ρ* was set to 1 and R = methyl was defined as the reference reaction (σ* = zero).  The factor of 1/2.48 is included to make σ* similar in magnitude to the Hammett σ values.