11. A definite numerical example of this kind of concentration of frequency about the mean was given in the note to §4. It was of a binomial form, consisting of the successive terms of the expansion of (1 + 1)^{m}. Now it may be shown (Quetelet, _Letters_, p. 263; Liagre, _Calcul des Probabilités_, §34) that the expansion of such a binomial, as m becomes indefinitely great, approaches as its limit the exponential form; that is, if we take a number of equidistant ordinates proportional respectively to 1, m, m(m - 1)/(1·2) &c., and connect their vertices, the figure we obtain approximately represents some form of the curve y = Ae^{-hx^{2}}, and tends to become identical with it, as m is increased without limit. In other words, if we suppose the errors to be produced by a limited number of finite, equal and independent causes, we have an approximation to the exponential Law of Error, which merges into identity as the causes are increased in number and diminished in magnitude without limit. Jevons has given (_Principles of Science_, p. 381) a diagram drawn to scale, to show how rapid this approximation is. One point must be carefully remembered here, as it is frequently overlooked (by Quetelet, for instance). The coefficients of a binomial of two equal terms--as (1 + 1)^{m}, in the preceding paragraph--are symmetrical in their arrangement from the first, and very speedily become indistinguishable in (graphical) outline from the final exponential form. But if, on the other hand, we were to consider the successive terms of such a binomial as (1 + 4)^{m} (which are proportional to the relative chances of 0, 1, 2, 3, ... failures in m ventures, of an event which has one chance in its favour to four against it) we should have an unsymmetrical succession. If however we suppose m to increase without limit, as in the former supposition, the unsymmetry gradually disappears and we tend towards precisely the same exponential form as if we had begun with two equal terms. The only difference is that the position of the vertex of the curve is no longer in the centre: in other words, the likeliest term or event is not an equal number of successes and failures but successes and failures in the ratio of 1 to 4. 12. 'Law of Error' is the usual technical term for what has been elsewhere spoken of above as a Law of Divergence from a mean. It is in strictness only appropriate in the case of one, namely the third, of the three classes of phenomena mentioned in §4, but by a convenient generalization it is equally applied to the other two; so that we term the amount of the divergence from the mean an 'error' in every case, however it may have been brought about. 13. This however seems to be the purport, either by direct assertion or by implication, of two elaborate works by Quetelet, viz. his _Physique Sociale_ and his _Anthropométrie_.