106 



Miscellanea 



and put a = 0, 6 = 1, c= - 1, rf = 2, e= -2, etc., we have 



„ (p-l)(j> + l)(y-2)(j> + 2)... ^ p(jo+l)(y-2)(;. + 2).. y(f)- 1)0>-2)( ;; + 2).. 



"'= (-l)(l)(-2)(2)... "»+(l)(2)(-l)(3)(-2)..."' + (-l)(-2)(l)(-3)(2)... "^ " 



=^^L«n + ,. JfL + ,., 'fzL + ... + ^^ + ...l (3), 



where it is assumed that 2,11 terms are used, so that 



;,,,„=(^ + 7i-l)...(i)-w) and o, = (- 1)»-' j^^^^l^^y^^. 



These two functions, p.>„ and e„ can be calculatod ejisily as in the csise of a two-variables inteiiio- 

 lation n will not he greater than 2 or for a one-variable thau 4. Taking ji = 4, < = 2 as an example 



_ 1 _ 1 

 ''~[2[5 ~240" 



For the two- variables we shall use (3) for obtaining an interpolated value in tho foUowing form 



and 





.(4). 



By this method we see the coiinection betwoeii Professor Everett's Central Diflerence* and 

 Lagrange's formula;, for if we write the differences in the foruier in terms of the function 

 and take p^n outside a bracket we see that the two are alike. In the actual work in the case of 

 a two-variables interpolation p.,„ x r.j„ could easily bo obtained and tlien we could find the 

 coefiBcients -i-^2nX''2n- Such reduced coefticient for i(, ., woukl be 



e,c, 



{p-t){r-s) 



Let US find as an example of the formulne an interpolated value for ages 51 and 28 from 

 the foUowing values at 3°/„ interest from the //" Joint Life Annuity Table t. 



' n au | (g + ^)''(g-^) a ,( g + 2)-(7-2 )„ , , (P-H)P(P-1) . (p + 2)...(p-2) 



where a^, a^... are cven central differences of ii,) and fco, b,... tho.se of i/,. — See Journal Institute of 

 Acluariei, Vol. 35, p. 452. 



t //■" is the nnnie given to the raortality table constructed by the Institute of Actuaries (lHfi9) from 

 the experience of licnltLy male lives assurcd by Knglish oflices. 



