The question of the premiums which should be charged to meet the eases of lives proposed for assurance admittedly incurring, or about to incur, risks more hazardous than those of the ordinary assured life, has from time to time engaged the attention of the Institute.

More stress has, perhaps, been laid upon the mortality of persons living in tropical countries, where the conditions of life are very different from those to which we are here accustomed, and statistics, obtained either from official sources or from business experience, have been furnished; whilst the only occupations to which attention has been directed are the Naval, Military, and Mercantile Marine Services, and those connected with the sale of alcohol.

With the view of obtaining, if possible, information respecting the modern practice of Life Assurance in connection with extra risks, I ventured to address to all the larger offices a Schedule intended to elicit such replies as would enable me to present a general idea of the methods applied to these cases.

In this I hope I have partially succeeded, but I need not add that the information must be regarded in a very general light. Many offices were particular to point out that, in the absence of any ascertained experience, each case of this nature would be considered upon its individual merits, and rated accordingly.

The main object of my paper on the Error introduced into Mortality Tables by Summation Formulas of Graduation, published at page 54 of the present volume of the Journal, has already been to a large extent fulfilled; and the admirable papers which it has called forth from Messrs. Lidstone and Spencer are a liberal reward for my labours. That object was, to show that the theoretical error in question is so small as to be quite insignificant; and thereby to bring summation formulas up again for discussion, with a view to ascertain which of them are the best as means of graduation pure and simple, and how they can be used most effectively.

In the concluding sentences of the paper I spoke of a graduation by summation, to be followed, in order to gain smoothness, by one by the graphic method; and in the reply to the discussion I casually threw out the idea of correcting the theoretical error by means of the fourth central difference; but these suggestions were only concessions to the misgivings of weaker brethren, and were not put forward seriously as being in my own opinion either necessary or desirable. I am convinced that we have now summation formulas which give as smooth graduations as need be desired, and the results of which conform with close accuracy to the original facts; and, emphasizing what has already been said, my own view is that the theoretical error they introduce is so minute as to be unworthy of consideration for a moment.