INTRODUCTION TO THE USE OF THE TABLES 



For this introduction to the use of the Tables I have largely drawn on the 

 prefaces to the original papers in Biometrika, and record here my acknowledge- 

 ments to the authors of the same. 



Interpolation. 



(1) A word must first be said as to interpolation. Let a function u be tabled 

 for the argument x proceeding by differences Ax = h. Then the scheme of such 

 a table with the differences of w is : 



a;_3 



ll-s 



■^—3 



?<_2 



a;_i 



W-1 



X(i 



u„ 



Xi 



Ml 



X.2 



11-2 



OS, 



u,. 



«4 



Ui 



•V, 



«5 



Aw._:, 



A-»_o 











A?/_., 



A=»._., 



Ah,_, 



Ah,_, 







Ai/_, 



A-»_ ; 



A^«_, 



Ah,_„ 







Ai(„ 





A»»_, 











A-»o 





Ah,_-, 



etc. 



etc 



A((, 



Ah,, 



A% 



Ah,o 







All, 



A-«., 



Ah,, 



Ah,, 







Aif, 



Ah,, 



Ah,, 









A», 













where : Aug = u^+i — Ug, 



Ahig = A?(s+i - Aug, 



A-'(/s = A-ifs^., — A^i,g etc., etc. 

 Now there are three interpolation formulae which it is desirable to remember. 

 If the function be required for the value x„ + dh and this value be termed Uo{6), 

 then we have : 



M„ {6) = Wo + dAu, - \, AX + -^^ ST ^ «« + 



2! 



3! 



■(i), 

 •(ii), 



«„(^)^«„ + ^A«„-^il^)A..-^i^)A=«_. 



where (f> = l — d. This is Everett's formula*. And lastly: 



«„ (e) = «„ +61 (A»„ + A«_0 + 1^ A^»_, - ^~~ i (A^iLi + A=«_3) . . . (iii), 



where we work with the differences on or adjacent to the horizontal through x„. 

 " Journal of the Institute of Actuaries, Vol. xxxv, p. 452. 



