284 PROCEEDINGS OF SECTION A. 



3. Mr. Makeham, as may be supposed, graduates by his modifi- 



cation of Gompertz's law. 



4. Mr. McKay's method claims to allow for the weight of obser- 



vations, and seems also based on some modification of 

 Gompertzs law. 

 After the adjusted table of mortality is obtained the calculation 

 of the annuities, assurances, and premiums at various rates of 

 interest becomes principally a nvimerical task in which the actuary 

 is aided by logarithms— ordinary and Gaussian — by tables of 

 interest, and by the help of the calculating machine, the arith- 

 mometer. 



There seems now little diversity of opinion that the method of 

 commutation columns is the best for obtaining the annuities, 

 premiums, and assurances, whether of, for, or on single or several 

 lives or survivorships, and they also give the simplest formulae 

 for short-term assurances, for increasing or diminishing assurances, 

 annuities, or premiums, return of premiums, endowments, endow- 

 ment assurances, &.c. 



For several lives these calculations become long and trouble- 

 some, and here the use of Gompertz's law (where applicable) very 

 much reduces the tables to be calculated, and, where not appli- 

 cable, the approximate formula? given by the application of the 

 differential and integral calculus for questions involving more than 

 two lives become indispensable. Possibly the simplest case of the 

 use of these commutation columns, i.e., a single life, may not be out 

 of place — 



D, = lif N, = 2 D, , 1 

 'V being the present value of £1 due one year hence. 

 C, = f4 V' " ' M, = S C, 

 TT^, the annual premium, = ^-'^— 



In chronological order, the next point where mathematics meet 

 the actuary is in the valuation of the assets and liabilities of a com- 

 pany, and ascertaining the profits or loss, and allocating (in a 

 mutual company) the bonuses. This, too, is best effected by the 

 commutation columns, or by the annuities deduced therefrom; for 

 instance, the best whole life formula is now considered to be 



.,v.= i-iT±i^" 



" •" 1 + Oj- 



Here, too, the question of interpolation comes in, to avoid the 

 necessit)^ of separate calculation of each value required of premiums 

 or what not, but every, say, fifth or tenth value is calculated, and 

 then the others are interpolated by the formidas derived from finite 

 differences of which those proceeding by central differences are 

 probably the best. Conic sections., and its extensions also here find 

 a place, as we find the actuaries now interpolating by means of 

 what they call the quartic parabola, whose equation appears to be 

 y ■=. ax •\- b x" ■\- c x^ -\- d x* 



