The suggestion has been made to me by Mr. George King, to write a paper upon an interesting problem which recently occupied our joint attention.

A brief statement of the case is as follows :

In a great public service there are certain well-defined ranks, the upper ones of which are exclusively recruited by promotion from the lower ones. There are limiting ages of retirement in each rank, and in the event of an officer reaching the fixed age without being promoted he is retired on a pension. The scale of pension depends mainly on length of service, varying of course with each rank, and, as the majority enter in the lowest rank between comparatively narrow limits of age, it is sufficient to deduce an average rate of pension at each age in the various ranks. The problem is to find what amount of annual contribution, uniform throughout future service, should be made in respect of each officer now in the service, of whatever age or rank, in order to provide for the ultimate charge for pensions.

In the particular case presented there were six ranks to be dealt with, and it was understood that the calculations were to be made at 3 per-cent compound interest. I trust I may be excused for this vague statement of the case, but I could hardly be more explicit without being too definite. I shall refer to these ranks throughout as first, second, third ranks, &c., where the first is understood to mean the highest.