There are few matters in connection with actuarial science of which it can be said with truth that professional knowledge has gained little during the past two decades—a period which has witnessed a remarkable expansion in both the theoretical and practical branches of the subject of life contingencies. Questions relating to extra risks, and the premiums which should be charged for them, scarcely appear, however, to have occupied the attention of actuaries to a degree at all commensurate with their commercial importance. Within the space of time alluded to above the practical side of the subject has received consideration at the hands of Mr. A. H. Bailey (J.I.A. xv, 77), and Mr. J. Meikle (J.I.A. xix, 268) ; while lines of inquiry and investigation bearing upon points of theory have been pursued by Mr. W. M. Makeham (J.I.A. xiv, 159 and 242), and Mr. J. R. Macfadyen (J.I.A. xvii, 77). Mr. A. G. Mackenzie has also contributed to our available stock of information on the subject in his Note on War Mortality, read before the Actuarial Society of Edinburgh in 1872. I venture to submit, however that there still remains much to be done to bring this branch of the main subject to the same state of enlightenment and completeness that other parts of the science have now attained. The following paper is an attempt to throw light upon a point which has hitherto escaped scientific treatment; and I entertain the hope that, in some cases of official routine, the methods proposed will be found of service.