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 u is : 



Am_ 3 



Att_„ 



A«_, 



A«„ 



A?/, 



Aw, 



Au., 



Au 4 



A*u_., 

 A 2 i/_ , 



A 2 «, 

 A 2 w., 



A 2 M, 



A 3 »_, 



A 3 «_. 2 



A 3 «_, 



A 3 «„ 



A 3 «, 



A 3 u„ 



A 4 «_., 

 A 4 '/_., 

 A 4 «_, etc., etc. 



A 4 M, 



vhere : 



A« g =»f s+1 - M„ 



A 2 */, = A« s+1 - Am,, 



A : 7/ s = AX+i - A 2 » s 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 u o (0) 

 then we have : ' 



«.(*)-«. + ^-^ AX+ ^Li^^^ + (i)> 



m^^^-Max-^5^., (ii) , 



where <£= 1-0. This is Everett's formula*. And lastly : 



«. (0) = «. + ^ | (A«„ + Aw.,) + t ^ _ ?Ojl^ | ( A 3 M _, + A 3 u_ 2 ) . . . (iii), 



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

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



