=The Origin.= The generating curve is the symmetrical normal probability curve with origin at its center. Since x = 0 when t = 0, the origin of the translated curve coincides with that of the base or generating curve. The translated curve may not be symmetrical so that the mean ordinate may not coincide with the modal ordinate. Because of the relation between corresponding areas the ordinate at the origin must continue to divide the area under the curve into equal parts, that is, the origin and median always coincide. =Determination of the Constants.= Since the exact position of the median can not ordinarily be determined by inspection or direct computation there are in reality four constants to be determined: the distance between the median and the mean, a, κ and λ. In determining the constants it is usual to compute the value of the first four moments. The third and fourth moments are extensions of the idea of the well known formulas for the first and second moments. Denoting the moments about the median by μ, we have 1 ⌠+∞ μ_{1}^´ = ——— │ xy dx N ⌡-∞ 1 ⌠+∞ μ_{2}^´ = ——— │ x^{2}y dx N ⌡-∞ 1 ⌠+∞ μ_{3}^´ = ——— │ x^{3}y dx N ⌡-∞ 1 ⌠+∞ μ_{4}^´ = ——— │ x^{4}y dx N ⌡-∞ where N is the total area under the curve. The values of the μ’s are computed from the data[8] and equated to the corresponding integrals which of course involve the four constants. In this way four equations are obtained from which the values of the constants may be determined. Since it is our present object to discuss the solution only of these equations, merely the principal results will be given. The general form for the moments about the median of the area under the translated curve is 1 ⌠+∞ μ_{n}^´ = ——— │ x^{n}y dx N ⌡-∞ 1 ⌠+∞ a^n(t + κt^2 + λt^3)^n = —————— │ ———————————————————————— e^{-t^2/2} a(1 + 2κt + 3λt^2) dt √(2π)N ⌡-∞ a(1 + 2κt + 3λt^2) 1 ⌠+∞ = —————— │ a^n(t + κt^2 + λt^3)^ne^{-t^2/2} dt. √(2π)N ⌡-∞ On applying the two well known formulas: ⌠+∞ │ x^{2n + 1}e^{-x^2} dx = 0 ⌡-∞ ⌠+∞ 2n + 1 ⌠+∞ │ x^{2n + 2}e^{-x^2} dx = —————— │ x^{2n}e^{-x^2} dx, ⌡-∞ 2 ⌡-∞ the determination of μ_1´, μ_2´, μ_3´ and μ_4´ is reduced to a matter of algebraic detail. Then on transferring to the arithmetic mean as origin the values of μ_2, μ_3, and μ_4 can be determined in terms of a, κ and λ. It is most convenient however, to make use of the quantities β_1 = μ_3^2/μ_2^3 and β_2 = μ_4/μ_2^2 or rather β = β_1/8 and ϵ = (β_2 - 3)/12 and express the constants in terms of these quantities. It is to be noted that both ϵ and β are zero for a normal distribution, that is, for λ = κ = 0. Omitting the detailed reduction[9] which is straightforward and direct, we have (1) μ´ = aκ