TRANSACTIONS OF SECTION A. 649' 



the best shape for practical calculatiou, ami which furnishes an intermediate step 

 in comparing the formula with Stirling's. 



To prove the formula, note first that since the second central difference A8f(.T) 

 0Tf{x + A.v) — 2f{.v)+f{.v~Ax) is not altered by changing the sign of Aa;, the 

 value of AS/(y) is the same when the independent variable is p as when it is q. 



Denote( ?^^"^---j |-[) by ct>^(q), so that <p„(q) = g, and cf>,iq) = (?^^J|(|=1) 



Then AS0r((/) = A8d)r-i(g). 



Making p the independent variable, and denoting the value of our series by 

 F (p), we have 



+pt/, + <j[>, {p)AdUj + <po(p)A:-h-u, + &c. 



ASF(p) = qA8i'„ + ({>Jq)A-8-Uf, + &c. 



+pA8u^ + 0|(/))A'-8-w, + &c. 

 A^b^Yip) = yA=-fiX + &c. +2)A'8''u, + &c. 



Giving jL» the values and 1, these become 



F(0) = Ko, •^5F( 0) = a8u,„ A'-S'F(O) = A'b'u^. 



F(l) = ?/„ a8F(1 j = a8u^, A'-S'-F(l) = A^d'ii,. 



Hence F (p) coincides with u for all the tabulated values that can be built up 

 from «o M, and their even central ditierences to the r^^ inclusive, that is for the 

 2r + 2 values of which Ug and m, are the two central. 



Moreover, when y is replaced by its value ] +p, F(j») has the standard 

 parabolic form 



A„ + A,/; + A,,/y' + . . . +A2r+ii>-'^+' 



containing 2r + 2 constants to be deduced from the given 2r + 2 tabulated values. 



Since A''8^0i.(a) = ^qC^') = ''> '^e may write <^r(.*) = ^-''d-''.i; it being understood 

 that in each inverse operation A-^ or §-% the arbitrary constant is to be so taken 

 as to make the result vanish with :v. The formula may then be written 



{q + (A6)-V . AS + (A^y-q . (.nS)- + &c.] M^ 

 + lp + (A8-> . AS + (a8)-> . (a8)- + &c.; Ml. 



In like manner the ordinary interpolation formula 



,i(.r— 1).., ., 

 H„ +.fAu^+ -A— ■'A-MQ+&C. 



can be written 



{1 + A-U . A + A--' 1 . A- + &c.} Ug-, 



and the formula for the same interpolanrt 



«o + .^■H + '^^,^ "t)''^ ^'% + &c. 



can be written 



{l + S-'l.S + 8--1.8- + &c.}«„ 



Taylor's Theorem can be written 



/(a + .(■) = ( 1 1- D-' 1 . D + D-- 1 . D- + &c.)/(«) ; 

 and the Binomial Theorem 



(1 + a)" = (1 + «A-> + a- A--' + &c.) 1, 



or 



(1 - rt)-x = (1 + rtS-> + a-8-- + &c.) 1. 



