METHODS OF INTERPOLATION. 341 



and their weights Ci, C2, &c., form two series the sum of whose indices 

 does not exceed 3, then the adjusted value which formula (97) gives is 

 not at all different from that which (69) will give when the intrinsic 

 weights are neglected. So, too, when the sum of the indices does not 

 exceed 5, we get the same result by using formula (95) and neglecting 

 the intrinsic weights, as we would get by taking them into account and 

 using formula (98). From this we may infer that when the intrinsic 

 weights form a regular sequence, it will make no great difference whether 

 we take them into account or not. But the more important point to 

 notice is, that the intrinsic weights of different terms do not bear to 

 each other the same relation as the weights of different observations of 

 the same term would do. For instance, in observing the value of Ui in 

 formula (98), any m observations, each of whose weights is n, are precisely 

 equivalent to any n observations each of whose weights is m; but these 

 m or n observations of u^ cannot be fully replaced by any number of 

 observations of Ui or W7, because those are quantities differing from 1(4 

 in magnitude, and their relation to % is but imperfectly known. This 

 consideration indicates that in making adjustments, the intrinsic weights 

 should be used ouly to a limited extent, if at all, and that the best we 

 can do is to observe the value of each term as accurately as possible, 

 and then regard all the terms as of equal weight, or nearly so. 



The same consideration will prevent our assigning different intrinsic 

 weights to the several terms combined together to form the sums Si, S2, 

 &c., in the first method of interpolation. 



ADJUSTMENT-FORMULAS OF OTHER WRITERS. 



Before quitting the subject of interpolation by the second method, the 

 present writer would observe that the process by which the new Insti- 

 tute of Actuaries' Life-Tables were adjusted, and which was first pub- 

 lished in the London Assurance Magazine for July, 1870, is really a 

 special case under this method, and by merely changing its notation, can 

 be reduced to the formula — 

 ^8 = .200 Us+.ld2 {u,-{- Wg) -1- .168 (We-f «io) +.056 (%-}- Un) +.024 (M4+ ^i^) 



+ .000 (1/3 + Uis) — .016 («2 + «i4) — .024 {til + M15) 

 Its publication was of later date than my first proposal of formulas of 

 this character for adjusting mortality-tables. The earlier portions of 

 my memoir, which contained formulas (13), (14), (15), and one or two 

 others of similar character, were presented to the Smithsonian Institu- 

 tion, and were sent by it to England for examination, in the summer of 

 1868. The formulas (17), (19), (20), &c., in which the weights form arith- 

 metical progressions, were presented later in the same year. When 

 these are extended so as to include fifteen terms, we get — 



Ws = j^ I 28 Us + 23 (M7 + «9) + 18 («e + -^(10) + 13 («g + Un) +8 (W4 + U12) 



+ 3 (W3 + «13) — 2 («2 + Wl4) — 7 («i + U15) \ 



and this is a better formula than the English one, as can be readily 



