upon the validity of Lanchester's Square Law as a model of attrition during the course of a battle. The solution of Lanchester's Square Law used here can be written as: a(t) = cosh(λt) - μsinh(λt) d(t) = cosh(λt) - (1/μ)sinh(λt) ε = λT where t is the time since the battle began, a(t) and d(t) are the surviving fractions of the attacker's and defender's forces at time t, λ is the Helmbold intensity parameter, μ is the Helmbold defender's advantage parameter, T is the duration of the battle, and ε is the Helmbold bitterness parameter. If the initial and final strengths of the two sides are known it is possible to solve for the parameters a(T), d(T), μ, and ε. If the battle duration T is also known, then it is possible to solve for λ If, as is normally the case, ε is small enough that the hyperbolic functions can, without any significant error, be replaced by their series expansion up to terms in the first power of ε, and if we adopt the following abbreviations for the casualty fractions FA = 1 - a(T) FD = 1 - d(T), then the following approximate relations hold: ε ≅ sqrt(FA * FD) μ ≅ FA / FD, That ε is a kind of "average" (specifically, the geometric mean) of the casualty fractions justifies using it as an index of the bitterness of the battle. We note here that for statistical work it is better to use the natural logarithms of the Helmbold Parameters. We will call them, in an obvious notation, logmu, logeps, and loglam. Major Findings See 1. The Helmbold parameters logeps and logmu are statistically independent, i.e., they measure distinct features of a battle. 2. The probability that the defender wins, P(Dwins), is related to the defender's advantage parameter via the logistic function, P(Dwins) = 1 / (1 + exp(-z)), with z = -0.1794 + 5.8694 * logmu. This logistic function is almost exactly skew-symmetric about logmu = 0, rising from P(Dwins) = 0.1 at logmu = -0.4, through P(DWins) = 0.5 at logmu = 0, to P(Dwins) = 0.9 at logmu = +0.4. Because the probability of victory depends on the Helmbold advantage parameter rather than the force ratio, it is clear that force ratio is an inferior and untrustworthy predictor of victory in battle. 3. While the defender's advantage varies widely from one battle to the