3 The only reason for supposing this exceptional shape is to secure simplicity. The ordinary target, allowing errors in two dimensions, would yield slightly more complicated results.
4 When first referred to, the general form of this equation was given (v.[ TN: space] 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.[ TN: space] Phil.[ TN: space] Trans.[ TN: space] vol. X.[ TN: space] p. 184, and vol. XIV.[ TN: space] 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.
8 The formula commonly used for the E.M.S. in this case is ∑e2/n − 1 and not ∑e2/n. The difference is trifling, unless n be small; the justification has been offered for it that since the sum of the squares measured from the true centre is a minimum (that centre being the ultimate arithmetical mean) the sum of the squares measured from the somewhat incorrectly assigned centre will be somewhat larger.
9 It appears to me that in strict logical propriety we should like to know the probable error committed in both the assignments of the preceding two sections. But the profound mathematicians who have discussed this question, and who alone are competent to treat it, have mostly written with the practical wants of Astronomy in view; and for this purpose it is sufficient to take account of the one great desideratum, viz.[** TN: space] the true values sought. Accordingly the only rules commonly given refer to the probable error of the mean.
10 i.e.[** TN: space] as distinguished from acting upon them indirectly. This latter proceeding, as explained in the chapter on Randomness, may result in giving a non-uniform distribution.
11 There is no difficulty in conceiving circumstances under which a law very closely resembling this would prevail. Suppose, e.g., that one of the two measurements had been made by a careful and skilled mechanic, and the other by a man who to save himself trouble had put in the estimate at random (within certain limits),—the firm having a knowledge of this fact but being of course unable to assign the two to their authors,—we should get very much such a Law of Error as is supposed above.