The method of calculating the values of prospective pensions which I am about to describe differs in some respects from those given in the standard papers on the subject, and will be found, I think, to possess advantages of its own. I shall deal with its application to the case of pensions based on salaries, when the amount of the pension is a uniform percentage of salary for each year of service taken account of.

I have followed Mr. Manly in the use of the following symbols (see his paper of 1901—J.I.A. xxxvi, 209), namely, lx to denote the number living and remaining on the active list at age x as given in the Service Table, and Dx and Nx to correspond; rx to denote the number retiring from failure of health between ages x and x + 1, sx to denote the salary receivable between ages x and x + 1, and Dsx to denote Dx X sx. When however lx, Dx, Nx, or Dsx have a circumflex accent, these symbols have the ordinary meanings, and the functions are based respectively on the Pensioners Mortality Table. I have found it necessary to employ also a certain number of new symbols which are described in the Key to the Special Notation (see pp. 31, 32). I have also followed Mr. Manly in the assumptions made in the paper mentioned above, that the retirements or superannuations applicable to any year of age take place at the end of the year, and that the pension commences at the same time and is payable yearly thereafter.