340 METHODS OF INTERPOLATION. 



intrinsic weight. This was attempted in formula (24) of the previous 

 memoir. We should now prefer to use one of the formulas of Table III, 

 rather than (16), and so write — 



^ _ .570 Cg U3 + .287 (C2 Uz + C4 ^^4) — -072 (ci th + C5 %) (nr,. 



^ .570 C3 + -^^^ (C3 + C4) — .072 (Ci + C5) V ^ 



The numerator here is formed by taking the series of products Ci Wi, C2 %, 

 &c., which are obtained by multiplying each given term by its intrinsic 

 weight, and adjusting them by formula ( 69). The denominator is formed 

 by adjusting the series of intrinsic weights alone with the same formula. 

 The aajusxfcd value of U3 is consequently exact whenever the series of 

 products and the series of intrinsic weights are both of an order not 

 higher than the third. But the product of two algebraic and entire 

 polynomials is of a degree equal to the sum of the degrees of the two 

 factors. Hence, formula (97)- will give exact results whenever the ob- 

 served terms %, th, &c., and their intrinsic weights Ci, C2, &c., form tv^o 

 series of such orders that the sum of their indices does not exceed 3. 

 Such will be the case, for example, when the former series is of the 

 second order, and the latter is of the first order, or an arithmetical pro- 

 gression. Generally speaking, there is no necessity that the intrinsic 

 weights should follow any sequence at all, but in the case of a table of 

 mortality they ordinarily do, the number of lives observed being, in a 

 rough way, a function of the age, as can be seen in the column of numbers 

 "Exposed to risk." at page 244 of the Mortality Experience already 

 referred to, the numbers increasing continuously up to age 40, where 

 they are at a maximum, and then diminishing continuously to the close 

 of life. If it would be too much to assume that any five consecutive 

 numbers in this series are in arithmetical progression, we can more 

 safely say that any seven of them form, approximately, a series of the 

 second or third order. IsTow, let us employ formula (95), which gives 

 exact results when applied to a series of the fifth or any lower order 

 Taking the intrinsic weights into account, we shall have — 

 ^ _ MOCiiii + .270{c3ih+C5U5)— ■'^OSjczth+CeUe) + ^OlS{ClUl-\-c^u^) ,q^. 

 ' .640c4+.270(C3+C5)-.108(C2 + C6) + .018(Ci + C7) ^ ' 



and this will be exact whenever the observed terms W], 112, &c., and their 

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

 does not exceed 5. Such will be the case, for instance, when the former 

 series is of an order not higher than the third, and the latter is of an 

 order not higher than the second, or vice versa. 



In applying this or any similar formula to the adjustment of a series, 

 such as (1) in Table IV, we can employ the known principle that the 

 intrinsic weights are inversely proportional to the squares of the mean 

 errors, and so take 



^ = (1001;) 



But there is reason to think that, in most cases, it is not best to take 

 the intrinsic weights into account. If the observed terms Ui, t<2, &c., 



