^{page 185 note *} Notable exceptions to this rule are Dr. T. B. Sprague's graphic method and the method of graduation with reference to a standard table. In the former method the operator is presumed to be influenced by his previous knowledge of the subject, but in a very indefinite not to say haphazard manner, and the latter method has only really been employed where data were limited.

^{page 185 note †} Other minor tests are also available.

^{page 186 note *} Hypothesis H is:—“That the true value which should have been “obtained by the observation u ₁ lies between u′ ₁ and u′ ₁ + σ, where σ is a small “constant number ; that the true value which should have been obtained by “the observation for u ₂ lies between u′ ₂ and u′ ₂ + σ and so on ; and finally the “true value which should have been obtained by the observation for u_n lies “between u′_n and u′_n + σ.” (Calculus of Observations, p. 304.)

^{page 186 note *} T.F.A., xl., p. 23.

^{page 188 note *} The mathematical basis of the calculation of mortality rates is discussed in T.F.A., viii., p. 163 et seq., by Prof. Whittaker, Mr. Lidstone, and others, but the point there at issue is not involved in the present inquiry.

^{page 188 note †} T.F.A., viii., p. 181.

^{page 189 note *} The late Sir G. F. Hardy suggested a curve of the form x^r (1–x)^s in this connection where the relative values of r and s depend on the most probable value of x (i.e. the value of q of which we are in search) and their absolute values on the extent of our prior knowledge of the function. T.F.A., viii., p. 181. While this curve yields a simpler process than that suggested in the Paper, it does not seem to rest on such a secure theoretical basis.

^{page 195 note *} T.F.A., x., p. 284.

^{page 195 note †} T.F.A., x., pp. 284-5.

^{page 214 note *} T.F.A., vol. x., p. 282.

^{page 216 note *} T.F.A., vol. viii., p. 163.

^{page 225 note *} Where exp (z) is a more easily printed form of e^z.

^{page 225 note †} Where the origin is θ and not q′ as in the authors' curve : i.e., x = q′ + z.

^{page 225 note ‡} The form (θ + r + 1)/(E + r + s + 2) given by Hardy gives the mean or “expected” value of q, as in Laplace's formula which Hardy was discussing: the above form gives the q of maximum probability, as taken by the authors in their own work, (Footnote continued from p. 225.)

^{page 227 note *} Sir John F. W. Herschel and Whewell were among the first to use neighbouring values to improve individual values (by means of a graphic process)—see J.I.A., vol. xxii., pp. 321–6, and vol. xxx., pp. 162–3.