The Head Injury Criterion (HIC) in FMVSS 208 for evaluating the potential head injury requires maximization of a mathematical expression, involving the time-average acceleration, by varying the limits (t1, t2) of the time interval over which the average is calculated.This paper describes the HIC behavior through the analysis of a function of two independent variables t1 and t2. The analysis is carried out for any arbitrary acceleration profile a(t). It is found that maximization requires that a(t1) = a(t2). Also, for the unique values of t1 and t2 that maximize HIC, the average acceleration between t1 and t2 is 5/3 times the acceleration at t1 or t2. Illustrative examples are provided by applying this condition to simple pulses. Numerical results are presented in tables and graphs.