VI.—On the Graduation of Data by the Orthogonal Polynomials of Least Squares

page 55 note * List A.

page 56 note * We would have used the notation x^[r] for this, but Steffensen has adopted that notation for what he defines as a central factorial, but what we should prefer to regard as a mean central factorial. Reference A (5).

page 59 note * Adopted by Tchebychef, Fisher, Jordan, Chotimsky, and Allan. References B (1), (2), (6), (11), (18).

page 65 note * According to precepts well established and justified. Cf. B 6 (a), (b), (c).

page 66 note * This example was constructed from a curve with inflexions, in order to bring out such a feature ; but the fact that the reduction in S² produced by the parabolic term a₂T₂ is almost zero was an unanticipated accident.

page 68 note * The expressions in central, not mean central factorials, were found in a different manner by Miss F. E. Allan, and more recently by P. Lorenz. References B, 1, 6 (c), and 12.

page 72 note * Unless one possesses a very efficient machine for ordinary repeated summation. By either mode of summation, ordinary or central, it is easy on a machine with a capacious register to sum two columns at once, and thus save the time and trouble of copying

page 75 note * Annals of Math., (2), vol. xxxii, 1931, pp. 461–462.

page 76 note * That it should not vanish is indeed the condition for the existence of the Legendre polynomials.

page 77 note * I am indebted to Professor J. F. Steffensen for the privilege of perusing his copy of this monograph.