In the present paper I propose to give some numerical illustrations of the working of the method of which the formulæ were given in the last Number of this Magazine, and to add some further results which have been subsequently arrived at.

In order to apply the formulæ to numerical cases, it is necessary to assume some law according to which the successive bonuses proceed. It is obvious, that whatever law is assumed must be to a great extent arbitrary, as it is of course quite impossible to say what will be the state of a particular Office after the lapse of a number of years. But it seems probable that, on an average, an Office may return to the assured in the form of bonus 20 per cent. of the premiums paid, and it is accordingly assumed in the calculations that each quinquennial cash bonus declared on a policy is equal in amount to one year's premium. This will probably not be far from the truth where the method of division of profits adopted in the Office is one that leads to nearly constant cash bonuses to each policy: such as the method proposed by Mr. Jellicoe, which divides the profits in proportion to the loading of the premium, accumulated at compound interest for the interval between two successive bonuses. But in cases where one of the more common methods of division is adopted, which lead to a series of increasing cash bonuses on each policy, the above illustration will not be applicable, and the conditions of the problem will be greatly changed. The calculations are made by means of the “Experience” mortality, assuming interest at 4 per cent.