[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.

the true values sought. Accordingly the only rules commonly given refer to the probable error of the mean.

[10] i.e.

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.

INDEX.