324 METHODS OF INTERPOLATION. 



similar way to find decimal weights for thirteen or more terms, as in 

 the following' cases : 



W7=.23466«:+.21137(W6+W8) + .14934(H5+W9) + .07003(w4+«io) ) 

 + .00105(M3+Mn)-.03005(«2+Mi2)-.01997(?fi+Wi3) ) 



^8= .20522 Us + .18953(2*7+ «9) + .14051 (^6+ "lo) + .08755(«5 + Wii) ) 



+ .02875(W4+Mi2)-.01321(W3+Mi3)-.02709(W2+Wi4) > (58) 

 -.01465(Mi+Wi5) ) 



In each of these formulas the sum of all the weights, taken for each 

 term separately, is unity, as it should be. Owing to the rejection of 

 decimals after the fifth figure, this condition would not always be ex- 

 actly satisfied, and consequently the fifth figure, as above given, has 

 been made to differ in some cases from its nearest value, to the extent 

 of a single unit of the fifth place. Actual trials have shown that a 

 better graduation can be made by these formulas than by any of the 

 similar ones previously given, and it is possible that, in some cases, a 

 table of mortality may be graduated sufficiently by this means alone, 

 without recourse to the first or third methods of adjustment. 



It will often be sufficient for practical purposes to use only three places 

 of decimals ; and in making an adjustment of a given series by any 

 single formula, we can facilitate the multiplications by preparing in 

 advance a table showing the product of each of the decimal weights by 

 each of the nine digits. 



There is another method, allied to the preceding, by which the 

 weights may be determined when more than five terms are to be included 

 in a formula. Supposing the number of terms to be seven, we may 

 assume that their seven weights, together with the two nearest zero 

 weights, are ordinates to a curve of the eighth degree, since such a 

 curve can be made to pass through nine given points. We have, as 

 before, the condition that this curve shall be tangent to the axis of X 

 at the points and 8 ; and to make its continuity with the axis at those 

 points as complete as possible, we may give it a contact of the second 

 order, so that its first and second differential coefficients shall both be- 

 come zero at the points and 8. We have thus the two conditions — 



'^^~'2 + 3"~4 + 5^~¥+T"'8-" 

 . ..,11J4 5J5. 137 Je 7 J7 , 363 ds_f. 



^^-^='+-12"- ir+T8(r— lo +-5Gr-" 



By means of these we obtain the two numbers — 

 3797C ,, 135087 



*'-li5r7 05ir 



and the adjustment formula is found to be— 



W4=ijJTTo[371712w4+236625(w3+?'r,) + 1^100(J<,+««)-32585(Wi + »7)J 



