300 METHODS OB^ INTERPOLATION. 



ages 100 to 108 can be computed by ordinary interpolation. Formula 

 (2) ma}- be used for this purpose. If we talce — 



Wl = «2 = W3^1, n=l, S^-«, Si=«i, S2=W2, S3=«3 



that formula reduces to — 



K=:«i«3(<,j + ^,3) V (32) 



If ?Ai, ?/-;,, W3, denote any three terms in a series, and the origin of coor- 

 dinates is at «2, and cii and «,-, denote the positive distances of Ui and 

 «3 from Uo. the above formula, enables us to interpolate any fourth term, 

 w, whose abscissa is x. If we now take — 



«i=l, «3=10, «i=37.oG3, W2=37.914, M3=:100 

 formula (32) becomes — 



tt=37.914+1.0G53^+ .51433 x^ 

 When the values 1, 2, 3, &c., areassigned to .rin this equation, the result- 

 ing values of M will be the desired terms for the ages 100, 101, 102, &c., 

 as they stand in column {h). The continuity of this added portion with 

 the rest of the series may be improved a little by adjusting, with form- 

 ula (20), a few of the terms adjacent to the point of junction. The ad- 

 justed values are as follows : 



A<:;e. 



96 35.535 



97 3G.30S 



98 37.139 



99 38.114 



100 : ... . 39.701 



101 42.180 



Series (/(), thus amended, is ready for practical use in the construction 

 of commutation tables. 



It is not claimed that this series is the best one which can possibly be 

 obtained by similar methods. The preliminary adjustment by the second 

 method admits of some variation, and repeated trials woidd be required 

 to determine Avhether the form of the final series might not be varied 

 with advantage by making it of some other order than the eighth, or by 

 taking the groujjs between some other limits than 10 and 99, or by both 

 these modifications together. But it is believed tliat the graduation 

 here obtained is accurate enough for practical purposes, and will com- 

 pare favorably with that of any table now in use. . 



We do not know, and perhai)s never can know, anything definite re- 

 specting the precise analytical form of that function which we call the 

 law of mortality. Various formidas, mostly transcendental, have been 

 devised to express it, but no one of them has yet received universal 

 recognition as correct to the exclusion of all otMers. While this state 



