page 298 note * It may not be out of place to append here another proof of Mr. Sprague's formula. Using and ¶_x and ¶_{x + t} in the extended sense given to them above, and supposing to mean the annuity payable from age (x + t) on to the expiry of the time named in the original contract, as also employing A _{x + t} to represent the single payment corresponding to the annual premium the amount of “paid-up” policy.

Divide both numerator and denominator by , and we have

But , and , therefore policy. Q.E.D.

page 300 note * The empirical method does not necessarily give greater result than the true ones, so we are quite entitled to suppose a case accordingly. Thus, represents the difference of the amounts by the two methods at end of one year; or the difference.

Now V _x|l is a maximum when the current risk is a minimum. Say, then, l_x =l _x+1. In which case , and , and the expression will become , or . But . ∴ our expression is , or . Now [except in the case when , or i = 0] n is greather than . Therefore, when the mortality is a minimum, the amount of paid-up policy at the end of one year, by the true method, will be greater than by the empirical method. Q. E. D.

page 301 note * Should the full value of policy not be allowed, but say, of it be deducted, and if some addition be made to the single payment, say, of it, then the amount of paid-up policy to be given would be .