METHODS OF INTERPOLATION. 



349 



the sum S of any group of n terms, the abscissa of whose middle point 

 is X. If we take — 



11 



^ = u 



we shall have — 



h = .09321G, I = .060463 



and consequently — 



u = 2.0259 /3^ + \ .89658 sin {x 0) - .01682 cos {x 0)] y'' 

 which may be transformed into — 



u = 2.0259 li^ + .89675 sin {xO-lo 4' 29") f 

 and this is the equation of the graduated series, the values of P, y] and 

 6 being, as already stated, 



)8= 1.0358, ;- = 1.0977, ^ = 3o27'37".8 



When the values — ^, +i, +f? &<''-j are successively assigned to x, the 

 resulting values of u will be the adjusted terms for the ages 54, 55, 56, 

 &c. The whole series is shown in the accompanying table. The sums 

 of the terms in the six assumed groups are equal to those in the given 

 series (/). 



Age. 



?{ 



Age. 



u 



Age. 



u 



Age. 



M 



Age. 



u 



10 



.4167 



28 



. 7210 



46 



1. 2958 



64 



3.977 



82 



16. 949 



11 



.4304 



29 



.7421 



47 



1.3531 



65 



4.310 



83 



18. 197 



12 



.4445 



30 



.7637 



48 



1. 4158 



66 



4.676 



84 



19. 497 



13 



. 4589 



31 



.7861 



49 



1. 4846 



67 



5.077 



85 



20.84 



14 



.4737 



32 



. 8092 



50 



1. 5601 



68 



5.516 



86 



22. 23 



15 



.4889 



33 



.8331 



51 



1. 6434 



09 



5.995 



87 



23 61 



16 



.5044 



34 



.8580 



52 



1. 7352 



70 



6.517 



88 



25.07 



17 



. 5202 



35 



.8840 



53 



1. 8366 



■ 71 



7.085 



89 



26.51 



18 



.5365 



36 



.9110 



54 



1. 9486 



72 



7.701 



90 



27.94 



19 



. 5531 



37 



.9395 



55 



2. 073 



73 



8.367 



91 



29.33 



20 



. 5701 



38 



.9695 



56 



2.210 



74 



9.087 



92 



30.67 



21 



. 5875 



39 



1. 0011 



57 



2.362 



75 



9.862 



93 



31.93 



22 



. 6052 



40 



1. 0348 



58 



2.530 



76 



10. 695 



94 



33.07 



23 



. 6234 



41 



1.0706 



59 



2.715 



77 



11. 586 



95 



34.06 



24 



. 6420 



42 



1. 1091 



60 



2.921 



78 



12. 538 



96 



34.86 



25 



.6610 



43 



1. 1503 



61 



3.147 



to 



13. 550 



97 



35.41 



26 



.6805 



44 



1. 1950 



62 



3.397 



80 



14. 624 



98 



35.67 



27 



.7005 



45 



1. 2432 



63 



3. 673 



81 



15. 7.58 



99 



35.57 



When we assume only three groups, Si, S2, and S3, in a given series, 

 the three constants will be — 



gh_ / S3 — S2 \ 



.(107) 



B = 



St So 



(/j^-l)(/3^'^i-^-*'^i) 



These are the same as in (67), except that the origin of co-ordinates is « 

 now placed at the middle of the series. 



The foregoing processes will serve for the purpose of ordinary inter- 

 polation from single terms, if we denote the given terms by Si, S2, &c., 

 and take — 



w = Wi = 7i z= 1 

 This corresponds to what has already been noticed in connection with 

 algebraic and circular functions. 



