[4] When first referred to, the general form of this equation was given (v.
[p. 29]). The special form here assigned, in which h/√π is substituted for A, is commonly employed in Probability, because the integral of y dx, between +∞ and −∞, becomes equal to unity. That is, the sum of all the mutually exclusive possibilities is represented, as usual, by unity. In this form of expression h is a quantity of the order x−1; for hx is to be a numerical quantity, standing as it does as an index. The modulus, being the reciprocal of this, is of the same order of quantities as the errors themselves. In fact, if we multiply it by 0.4769… we have the so-called ‘probable error.’
[5] See, for the explanation of this, and of the graphical method of illustrating it, the note on [p. 29].
[6] Broadly speaking, we may say that the above remarks hold good of any law of frequency of error in which there are actual limits, however wide, to the possible magnitude of an error. If there are no limits to the possible errors, this characteristic of an average to heap its results up towards the centre will depend upon circumstances. When, as in the exponential curve, the approximation to the base, as asymptote, is exceedingly rapid,—that is, when the extreme errors are relatively very few,—it still holds good. But if we were to take as our law of facility such an equation as y = π/1 + x2, (as hinted by De Morgan and noted by Mr Edgeworth: Camb.
Phil.
Trans.
vol. X.
p. 184, and vol. XIV.
p. 160) it does not hold good. The result of averaging is to diminish the tendency to cluster towards the centre.
[7] The reader will find the proofs of these and other similar formulæ in Galloway on Probability, and in Airy on Errors.