In 1869 Mr. Sprague read before the Institute an interesting paper “On the value of Reversionary Annuities payable half-yearly, quarterly, &c., according to the conditions which prevail in practice.” This valuable paper is given in the Journal, vol. xv, page 126. The conditions which prevail in practice, as here for the first time introduced and mathematically investigated by Mr. Sprague, are the following:—For a yearly reversionary annuity on a life x after the death of another life y, the first payment of the annuity is supposed to be made just one year after the day of the death of y, provided that x be then living; and for a like annuity payable by m instalments in each year, the first payment of an instalment is supposed to be made just of a year after the day of the death of y. Also, in each case, it is further supposed that whenever x shall die, after surviving y, the annuity shall be completed up to the day of the death of x by the supplementary payment of the proportion of the annuity estimated in respect of the fractional interval that may then have elapsed since the last yearly payment or instalment was received.

The following results compiled from my paper “On the Progress to Accuracy ofLogarithmic Tables,” read before the Royal Astronomical Society, on 13thDecember 1872, will probably be found interesting to the readers of the Journal of the Institute of Actuaries, more especiallyin connection with the recent correspondence in reference to Mr. Sang'slogarithmic tables.

If a classification be made of any considerable number of life policies according to age at issue, a cursory inspection will show that the numbers at the different ages are not fortuitous, but are subject to some unknown, though plainly marked, mathematical relation; and the amenity of the results to law will be more evident as the number of policies employed in the classification is larger, and the effect of merely accidental fluctuations is consequently more nearly eliminated.

Besides the probable error above considered, there is another quantity ofsome importance in practice, which may be called the Risk of a departurefrom the most probable number of survivers. It is obtained by addingtogether the products of all the possible positive deviations in the numberof the survivers from its most probable value, each multiplied into theprobability of its occurrence; or by adding together the products of all thepossible negative deviations in the number of the survivers from its mostprobable value, each multiplied into the probability of its occurrence; inthis case, however, taken without sign. For if we consider the deviationsfrom the most probable number of survivers, taken without regard to sign, asrepresented by proportional sums of money, then the quantity just definedevidently expresses either the mathematical expectation of gain of a personwho is to receive those sums, or the mathematical expectation of loss of aperson who is to pay them. The latter of these two ideas more generallyoccurs in practice.

It is well known that the number of the physical sciences is continually increasing, by a process of development from within. We constantly see divisions and branches of existing sciences cultivated and developed until they acquire a separate existence as new and independent sciences. For it has long been impossible for a single person to master the whole, or even a considerable part of these sciences, and the necessity for a division of labour is continually becoming more and more felt here, as elsewhere. It is, however, comparatively seldom that the number of the sciences is increased by the addition to them of an entirely new one, and therefore the case of which I am about to speak appears to deserve our particular attention. For I shall presently prove that a department of science, which has not hitherto belonged to the physical sciences, is now being so developed and perfected, that in a very short time it will claim admission into the number on equal terms with the others. This department of science is Statistics. The science of Statistics in its present form has now existed a full century. It owed its birth, as well as its name, to Achenwall of Göttingen, about the middle of the last century.

Never was a more complete change suddenly brought about in the laws of a nation by legislative enactment than that which has taken place this year in France in the law of life insurance, in consequence of one single decision of the Supreme Court of Judicature, Up to the present time the construction given to the contract of life insurance in this country has been quite different from what it is in England. As there is no provision of written law that relates to life insurance, it being not even so much as mentioned in the Civil or Commercial Codes, people thought themselves justified in governing it by laws and rules of their own. For instance, although it is a principle of law common to both English and French jurisprudence (we may add, to the law of all legislating nations from time immemorial) that choses in action shall necessarily devolve upon our legal representatives after our death, it has hitherto been decided almost universally by French tribunals that an exception was to be made in favour of life insurance policies. By the advocates of that doctrine, the right in the sum assured was thought never to have vested in the person effecting the policy, and the assurance monies were said to be transferred directly, i. e., omisso medio, from the assurer to the party entitled to receive the sum assured; and that sum, accordingly, would not be liable to succession duty.