47() Tahles of Poivers and Sums qf Powers 



Illustration II. 



To show tili' usc of tlic secoml table wo will liiiil, l)v iHl' inetliod of least 

 Squares, a parabola of the third oider fitting the foUowiiig expectatioiis of life 

 which have bcen taken froin the recent experience of fcmale aiinuitants collected 

 by the Institute and Facuity of Actuaries (C. and E. Laytoii, 1899). 



I took age 15 as origin and calculated the products of the expectations into 

 the first three power.s of the distances from thi.s age -=- .5 continuously am! checked 

 the work by multiplying the expectation by the appropriate value of »' from 

 Table I. 



Then since the type-equation for a, is 



we can write down the equations for fiuding the as immediately by using Table II. 

 with )i = 1 7 and we have 



17o„+ löSa, + 178.5«,+ 23409(;3= 328-6 



153a„+ 178.5ai+ 23409«.,+ 327369«,= 1782-6 



1785ao+ 23409«,+ 327369«,,+ 4767633«,= 141.52-8 



23409a„+ 327369«! + 4767633«.,+ 7 1397705«3= 139461-0 



The Solution of these equations gives 



expectation of life = 47-46 - 2-6013a' - -18435a,-" + -01063a;' 



and WC obtain the tiillowing valucs agreeing very closely with the unadju.sted 

 expectations : 



