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Published online by Cambridge University Press: 15 September 2014
The problem of Graduation or Adjustment, with which the present paper is concerned, may be defined as follows. Let a number u be a function of a number x: and suppose that, corresponding to the values … −3, − 2, −1, 0, 1, 2, 3 … of x, we have obtained, as a result of observation, values … u−3, u−2, u−1, u0, u1, u2, u3 … for u. Owing to errors of observation, these observed values when plotted against the corresponding values of x do not lie on a smooth curve, although for theoretical reasons we believe that they would do so if freed from errors. The problem is to determine the most probable set of “graduated” or “adjusted” values
which differ only slightly from the above observed values, and which lie on a smooth curve.
page 112 note * Journ. Inst. Act., 22, p. 320, and 30, p. 162.
page 113 note * Journ. Inst. Act., 38, p. 334.
page 113 note † Usually small multiples.
page 114 note * Proc. of the Vth Internat. Congress of Mathematicians, Cambridge, 1912, ii, p. 348; Proc. of the London Mathematical Society, ser. 2, vol. xiii, part ii; Journ. Inst. Act., 48, p. 171; 48, p. 390; 49, p. 148.Google Scholar
page 116 note * I must express my obligations to Mr John Maclean, M.A., B.Sc., Professor of Mathematics in Wilson College, Bombay, and to Mr Jason J. Nassau, C.E., M.Sc., Instructor in Mathematics in Syracuse University, N.Y., U.S.A., who took a great share in performing the computations, which were carried out in the Mathematical Institute of Edinburgh University. It should also be mentioned that Mr W. F. Sheppard (loc. cit.) had previously discussed the cases k = 0, 1, 2; but his formulæ, being expressed in terms of c0 (Sheppard's b0), ∂2c0, ∂4c0 … in one paper and in terms of central sums in another paper, are not the same as the formulæ found above under the headings k = 0, 1, 2, although, of course, they may be shown to be equivalent to them.
page 118 note * Journ. Inst. Act., 41, p. 348. See also Karup (J.), Trans, of 2nd International Actuarial Congress, p. 47, and translation, p. 93.
page 119 note * As is easily seen by expanding the elements in the first row of the numerator determinant in equation (A).
page 121 note * See also Trans. Act. Soc. of America, vol. xix, pp. 302-3.
