348 METHODS OF INTERPOLATION. 



number of groups was 2m + 1, the series will be of the (»} + l)th order, 

 but after the constant A has been subtra.eted from each term, the 

 remainders will form a series of the mth order. 



We will now make an application of this method to the graduation 

 of series (/) in Table II of the previous memoir, in order to illustrate the 

 working of the formulas, but without any design of specially recom- 

 mending the method for adjusting mortality-tables. Let us assume six 

 consecutive groups of equal extent, so as to have — 



m = 3, Wi = /«, = 15 



The sums of the terms in these six groups are — 



Si= 7.8522 83 = 21.069 85=182.02 



82 = 12.255 84 = 54.573 Sg = 446.84 



Substituting these values in the equations (103), we have three equa- 

 tions from which we find the three-scale terms — 



Ai = - 27.799, A2 = 24.863, A3 = - 6.6914 



and the equation of relation therefore is — 



^3 _ 6.6914 z^ + 24.863 z - 27.799 = 

 This has only one real root, 



z = ^^^= 1.6960 ; 

 and consequently we have /5 = 1.0358. Dividing the equatioa of rela- 

 tion by — 



z — 1.6960 = 



we get the quadratic factor — 



«2-4.9954;s-f 16.391 = 



which contains the pair of imaginary roots. Hence, by the formulas 

 (105) we have — 



16.391 = f^^ 4.9954 = 2 y^^' cos 15^ 



which give for the values of the two constants — 



J- = 1.0977, ^ = 30 27'37".8 



!N"ow, substituting the values of /3, y^ and e in (102), and taking — 



n=ini = 15, A = 



we find — 



Ti = 1.3623, I = 1.0981 



and assigning to 8 in succession the values of 84, 85, and 85, and to cc 



the corresponding values J^% -Y-, and -^/, we get the three equations — 



54.573= .69599 B+ 3.1862 C + 1.4977 D 



182.02 = 1.1804 B + 12.731 — 6.4117 D 



446.84 =2.0019 B + 11.368 C — 56.579 D 



from which the values of the three remaining constants are found to be — 



B = 57.520, C = 6.6876, D = - 4.5182 



We have thus determined all the constants of formula (102) which are 

 required for the case in hand, and are enabled to find by interpolation 



