334 EEPORT— 1880. 



be the coefficient of a;'' in the binomial expansion of (1 + a;)^""\ the 

 coi'responding formula is 



it,. 



(k^ - Z'-) (9 P - ^2) (2„_02 P - ^2 



2 . 4 . 6 (4^-2) 7c*''-3 



^>» A _^ t>„ Ai _ i'„_i B i'„-i Bi _^ 

 h + z Ic — z 3^—2 3 Zv + 2 



Thus if n=l 



(2)i-l)Z;-2 - (2?i- 1)Z; + z 



\ 



77 — i^~^A j-I-A+Ja 



^x — 2 ~7~ -"- T^ 2 7 -^1 



= A[ when z ■■= + h 

 = A -when z 



_ 7. f (as ifc ought). 



This formula gives a very important theorem for the bisection of aji 

 interval. Making 2 = 0, Zc divides out, and there remains 



12.32.52 (2»i-l)2r ,4^.. g 



2 . 4 . 6 ...... (4 91 — 2) L 



I tV_i (B + BO + ^ t'„-2 (G+GO- } 



When 91 = 1 , 2 Mo = A + Ai 



n = 2,l6 ?(o = 9 (A + Ai) - (B + BO 



w=3,256«o = 150(A + A0-25(B + BO + 3 (C + CO- 



The general case of n = 1 is simply equivalent to the use of pro- 

 portional jDarts. 



!0 Although, as has been already remarked, the rules for quadrature by 

 ordinates can be obtained by integrating the corresponding expressions 

 for interpolation by ordinates, that is not the easiest way of obtaining 



them. One way is to integrate / (1 + A)"^ dx after expanding it, and 



then, rejecting all the terms after A*", make n = r and substitute E — 1 

 for A. This process is given in most of the text books. f 



But a simpler and more symmetrical method, and one which can 

 easily be extended to higher integrals, is by the use of indeterminate 



multipliers, as follows : Write 21 = Uq + a, x^ + a^x* + ai^x*^ + 



whence 



1 /^" 111 



- / icdx = naQ+ ~n^a2+ --n^a^ + ^ii^ag + • . . . 



2 ^-n 6 o i 



Again substituting 0, i 1, ± 2, + 3 in succession for x, 



Uq = Uq 



1 



(«-l + Ml) = ao + ^2 + «4 + «6 + 



* See De Morgan, I>ijf. and Int. Cede. p. 5i9. 



I See Boole, Finite Differences, art. 10 ; also Murray's Shijjhuilding, p. 32. 



