^{page 215 note *} We remind the reader of a very closely allied method of investigation applied in the calculus of probabilities.

^{page 218 note *} Since the above was written, I have become aware that my deduction is not quite correct. Different causes of death, although they may be quite independent of each other, constantly disturb one another; and their joint effect is not equal to the sum of effects of every cause taken separately. On the other hand, the analytical expression of a cause of death acting concurrently with other causes, differs from the expression of the cause acting alone. This ought to have been taken into consideration before I tried to explain the constants of the formula; v and w being the causes, each acting concurrently with the other.

Nov. 1873.

W.L.

With reference to this, a correspondent writes.

Does not the modification of his hypothesis, which Mr. L. here finds himself constrained to admit, strike at the root of his whole argument ? In Vital Statistics, as in other branches of Social Science, in Physiology, and even in Chemistry, it is not true, but very far from being true, that the effects of causes acting in combination are equal to the sum of the effects of the same causes acting separately. But if this is so, must we not regard Vital Statistics as a strictly experimental or empirical science, the complex and often fugitive phenomena of which cannot be legitimately brought within the boundaries of formulæ which are applicable to dynamics or the strictly mathematical branches of physics?

Mr. L. takes as an illustration of the value of his hypothesis, drawn from another branch of science, the Epicycles of Ptolemy. But were the epicycles really a help or a hindrance to the progress of astronomical discovery ? Did they not rather stand in the way of true investigation, unti l their increasing number and complexity showed clearly that they did not answer to anything in nature; but were mere arbitrary hypotheses invented to cover facts which they did not truly explain, and they gave way before the simple and natural theory of Copernicus? Without absolutely denying the value of a temporary hypothesis—clearly recognized as such—as a mode of arranging and classifying facts, it is scarcely necessary to say that any practical use of it in such a field as Vital Statistics might prove misleading and hazardous.

W. S.

^{page 219 note *} Let r denote the period, then the probability of living thro' it is . The logarithm of this, l_x+1 −logl_x , is easily seen to be equal to c + q^x+1 logk+(x+1)log h-c-q^x log k-xlog h =q^x (q^r -1)log k=rlog h; and if we add to this the constant quantity r log =-rlog h, we get q^r(q^r-1)log k, the values of which for equidistant values of x, form a geometrical series.

^{page 220 note *} Disregarding the mortality of the first period, the constants in the formula may be approximately determined from the mortality table as follows:—

In the same manner we find

Forming now the second differences, we get

and dividing the latter of these equations by the former,

whence q can be immediately determined, and with the greater accuracy the greater n is taken. We require, for this purpose, to take four equidistant terms of the mortality table, l_x, l_x+n, l_x+2n, l_x+3n If we put

also ,

then .

From the same four values of the mortality table h is most simply determined as follows:

Consequently ,

and .

The determination of k and c presents no difficulty.

^{page 223 note *} The constants in this case are, if we start with l₂₀, l₄₀, l₆₀, l₈₀ , and use the uncorrected observed numbers, so that

To complete the table at the younger ages, I put