^{page 164 note *} Bayes, phil. Trans. 53 (1763), p. 370; Laplace, Mém. par divers Savans 6 (1774).

^{page 164 note *} “Both from the point of view of practical common-sense, and from the point of view of logic, the so-called laws of Inverse Probability are a useless appendage to the first principles of probability, if indeed they be not a flat contradiction of those very principles.…The laws of Inverse Probability being dead, should le decently buried out of sight, and not embalmed in text-books and examination papers.”—Chrystal, Trans. Actuarial Soc. Edinb 2 (1891), p 421.

^{page 164 note †} loc. cit.

^{page 168 note *} is used to denote

^{page 170 note *} Tidsskrift for Math. (4) 3 (1879) pp. 1, 66, 122.

^{page 171 note *} Insurance Record, Oct. 25, 1889, p. 433.

^{page 174 note *} This summary is given with the kind permission of the Proprietors of the Insurance Record.

^{page 179 note *} [It will be observed that the Reviewer's solution differs from that given by Professor Whittaker in § 6 of his paper. The difference is due to their formulating in two different ways (both legitimate) the state of complete a priori ignorance. The Reviewer considers the hypothesis that (1) the facility for surviving a year at age k lies between x and x + dx, and (2) the facility for surviving a year at age (k + 1) lies between y and y + dy; and he assumes, that the a priori probability of this hypothesis is proportional to dxdy. Professor Whittaker, on the other hand, considers the hypothesis that (1) for a man aged 70 the facility of death between his 70th and 71st birthdays lies between x and x + dx and (2) for a man aged 70 the facility of death between his 71st and 72nd birthdays lies between y and y + dy; and he assumes that the a priori probability of this hypothesis is proportional to dxdy.]

^{page 182 note *} This treatise is commonly bound up with Vol. II. of Jones on Annuities, and is there accessible to actuarial students; but the demonstration there given is rather condensed and the notation needs careful attention.

^{page 183 note *} The transition from finite summation to integration seems to involve the tacit assumption that B, the number of balls in the bag, is theoretically infinite, or practically so large that there is virtually no–distinction between the value of the sum and that of the integral. For example, B = 10, if the possible values of the x's are confined to proper fractions with zero or integral numerators and the denominator 10, i.e. they are not capable of continuous variation, but only of variation by finite steps of

^{page 183 note †} See footnote to § 15.

^{page 184 note *} The final results (7), (8), and (9) will show that, as must of course be the case, the order in which we consider the variables is immaterial.

^{page 186 note *} See the discussion of this question in § 7 of Professor Whittaker's paper in the simple case of two repetitions.

^{page 187 note *} I am indebted to Professor Whittaker for calling attention to the considerations here referred to, which are of fundamental importance for a correct understanding of the subject; cf. Edgeworth, Encyc.-Brit., 11th ed., vol. xxii., p. 377c. “What is the probability of drawing a white ball from a box of which we know only that it contains balls both black and white” [or rather, either white or black], “and none of any other colour? In this case, unlike the case of an urn containing a mixture of white and black balls in equal proportions, we have no reason to expect that if we go on drawing balls from the urn, replacing each ball after it has been drawn, the series so presented will consist of white and black in about equal numbers. But there is ground for believing that in the long course of experience in pari materia—other urns of similar constitution, other cases in which there is no reason to expect one alternative more than another—an event of one kind will occur as often as one of another kind.” This, of course, is the simpler case, in which all our m's ≡ 0, i.e. we have no a posteriori evidence of the actual constitution of the urn, but the principle is the same.—G. J . L.

^{page 188 note *} This is used, as in § 4, as an abbreviation of “respectively lie between x ₁ and x₁ + dx₁, x₂ + dx₂,” and so on.

^{page 188 note †} On the hypothesis of § 2, U is a constant = i – 1; and it will be found that the a priori chance of the proportion of balls of any specified colour being x is and the average value of as previously found.

^{page 190 note *} It is tacitly assumed that no two of the points are coincident. This is justified by the prior assumption—see footnote to § 3—that the total number of balls tends to infinity, so that in any finite number of drawings the chance of coincidence is infinitely small.

^{page 191 note *} It may possibly be open to question whether Crofton's process, here followed and extended, ean be considered strictly demonstrative; but it is at least simple, elegant, suggestive, and easily carried in the mind.

^{page 196 note *} This is the condition that shall be finite, which in necessary since we must have

^{page 197 note *} Vol. III. part i. page 141. Read 26th May 1828.

^{page 197 note †} These are applications of the formula (m + 1)/(Σm + i), where i is here the number of unit-intervals in 10 years, increased by 1 to represent the case of survivance); see § 9, p. 185.

^{page 199 note *} If t is the unit of time, the probabilities produced by the formula, as used by the Professor, are of the form q, _t|q, _2t|q, …, _tp, not ^oq, ^tq, ^2tq … p.

^{page 201 note *} In a subsequent note to the Editor, Mr. Armstrong says: “I venture to emphasise the distinction between as the crude or ungraduated value of an isolated probability of death and the value of the latter as one of a series in the final graduated Mortality Table. The graduated values stand for the probabilities that would, so far as we know, have resulted from data indefinitely more extensive. This is what formula (3) (p. 169) also aims at. There the corrective function v(x) is assumed to be given by previous investigations. The Actuary, on the other hand, confines himself to the experience of a particular group over a specified period, and “corrects” the crude probability at each age by making use of those at adjacent ages, weighted more or less according to the graduation formula employed. On this view the controversy as to the true or theoretical value of q_x would seem to resolve itself into a question of the proper mode of graduation.”

^{page 204 note *} Vide §§ 17–20 of Mr. Lidstone's Note, ante, pp. 190–192, which sections were added after the date of the Discussion, and before receipt of Mr. Nicholl's remarks.