The subject of sickness insurance does not appear to have been previously discussed before this Institute except in relation to Friendly Societies and the National Health Insurance Act. There is, however, a most comprehensive paper by Mr W. A. Robertson, the present President of the Faculty, in the Transactions, Vol. xiv, page 21. Disability in the sense of long-term sickness insurance has been considered on several occasions, but purely as an added benefit to life assurance contracts. The results shown by the American offices, who at one time transacted a large volume of this business, have been anything but encouraging.

If a pebble on a steep hill is disturbed from a state of rest, it will probably roll down the hill gathering speed as it goes, and, if the hill ends in a precipice, the pebble will, unless it is checked in its course, eventually fall over the edge. So long as the pebble is rolling down the hill it is within sight of an observer on the hill, but when it falls over the precipice it passes out of his sight; its disappearance occurs suddenly. There is no violent change in the pebble's speed or direction as it passes over the edge, so the suddenness of its disappearance is not due to any alteration in its motion but to the fact that it has rolled down beyond the level of the edge of the hill.

The recent paper by Dr Hagstroem (J.I.A. Vol. LXX, p. 119) directs attention to the very closely related subject of the effect on life assurance premiums of changes in the rate of interest. Some four years ago, in the course of an address before the American Life Convention, Mr V. R. Smith studied the relationship of the interest rate to life assurance premiums from a different angle. This note is an attempt to develop the mathematical theory underlying Mr Smith's method and to present some results on the basis of the A 1924–29 Table.