where the differences constitute series of rapidly diminishing terms. The only term of the second order which this addition to the hypothesis introduces into —J ~t 't.dt is aA2j3 f (t — z) dt which is =0, when taken from t=0 to t= 1. Consequently the errors of the rule are all of the third order.
To give a notion of the amount of error, extend the preceding formula: to terms of the third order, and form the integral, reserving only the terms of the third order. The final result is as follows : — If x and y be the number of the living at the age of A and B, and if a, b, c, .. . he the numbers alive, at n, n + 1, ... years older than A, and p, q, r, ... at n, n + 1, ... years older than B, then the probability that A shall begin to survive B in the course of the (n + 1) th year of the calculation, is
(a+b)(p—q) (b—e)(p—q) - (a—b)(q—r)
2 x y + 12xy
the first term being that generally used, and the second a correction which ought always to be applied in those parts of the table in which the yearly decrements are not equal.
The demonstration of the preceding will be easily arrived at by the indication which I have given, by any one acquainted with the integral calculus. To those who have not that advantage, reason may be shown in the result, though not for the result. The preceding
XXIV APPENDIX THE FOURTH.
correction will have the positive sign when b—c bears a greater proportion to a — h than does q — r to P — q : that is, when the mortality in A's part of the table is in-creasing faster than in B's. Now, ceteris paribus, the larger the comparative mortality of the year succeeding a given year, the more likely are the deaths of the latter part of that given year to predominate over those of the former; consequently, the more likely is the death of A, if it happen in that year, to be towards the end of it. But any thing which shows that the death of A is more likely than before to take place later in its year, increases the probability that a survivorship commencing in that year shall be in favour of A, and not of B.
APPENDIX THE FOURTH.
ON THE AVERAGE RESULT OF A NUMBER OF OB
SERVATIONS.
THAT I might not further embarrass the most abchapter of this work, by the introduction of an isolated point of difficulty, I have chosen here to men_ tion some considerations connected with the value of the average of observations. There is a remarkable difference of principle between two problems which at first sight appear identical; namely, where it is required to invent a method of treating observations be-fore they are made, and after they are made. Positive and negative errors being equally likely, and no observhaving been made, it is easily proved that there is a high probability in favour of a large number of observations giving exactly or nearly the same total amount of one as of the other. The case is analogous to that of an urn filled with black and white balls in
ON THE AVERAGE RESULT OF OBSERVATIONS. XXV
equal numbers, out of which, in a large number of drawings, both sorts will come in nearly equal propor
But in this, as in every other question of probaany additional knowledge of the circumstances which may happen, or have happened, changes the problem, and is equivalent to an extension or limitation of its conditions. When the observations have been made, the position of the observer is altered, since though the law of facility of error be not determined, yet more probability is given to some laws than to others, by inspection of the observations themselves. For instance; if the observations give results of very little discordance, it is immediately obvious that a law of facility which makes the probability of large errors very small, is more likely to have been that which actually existed than one of a different character. The problem now presents an analogy with that of an urn, from which drawings have been made and registered, so that the contents of the urn are to be guessed at from the drawings.
In the first problem, and supposing that a method of combining the observations is to be chosen before obsermade, it is demonstrable that the average of the results is more likely to be true than any other magnitude. And the same conclusion seems probable in the second case, since unassisted common sense would never draw any distinction between the two problems. But the results of calculation applied to the development of the distinction just drawn, show that the average of obsermade is not necessarily the most probable renor can be such for more than two observations, unless one particular law of facility of error be sup-posed, which law is the standard law described in Chapter VII. But it is also shown, as mentioned in page 142, that the results of any law of facility, when applied to tolerably large numbers of observations, are nearly identical with those of some variety or other of the standard law; so that, practically, the average of
xXVi APPENDIX THE FIFTH.
observations is either the result which the strictest application of sound principles would declare to be the most probable truth, or else very near to it.
It is, in the meanwhile, a most remarkable circumstance that a method so simple, and so conformable to common sense, as that of averaging, should first turn out to be incorrect, except upon a supposition never contemplated in thinking of the evidence of this rule, and should after-wards prove to be always nearly correct, for large numof observations, on account of the tendency of all admissible suppositions to confound themselves, as the number of observations increases, with that one particular supposition, which makes the common notion absolutely correct. My own impression, derived from this and many other circumstances connected with the analysis of probabilities, is, that mathematical results have outrun their interpretation : and that some simple explanation of the force and meaning of the celebrated integral, whose values are tabulated at the end of this work, will one day be found to connect the higher and lower parts of the subject with a degree of simplicity which will at once render useless (except to the hisall the works hitherto written.
APPENDIX THE FIFTH.
ON THE METHOD OF CALCULATING UNIFORMLY DE.
CREASING OR INCREASING ANNUITIES.
AN authority from which I rarely differ has spoken thus, " A few writers on these subjects, of late years, have employed the differential and integral calculus in their investigations. We have not yet seen any fruits