-1 



354 « REPORT — 1880. 



rt, = a + ( B» + -^ hxA 



+ (2Cx + i cx^) X ^^ 



+ (5E. + - ex^) X 3-^. ^_g-. ^-^ + &c. 



the common interval being supposed unity. If the interval is other than 

 unity, say h, the formula must be rendered homogeneous by the substitu- 

 tion ot X : h for X. 



For the formula symmetrical to an interval, the differences run 



'?^— I • • • * ^ ""—2 • • . • -^ '^—3 • • • • 



Am_, .... A%_2 .... A5tt_3 &c. 

 ^^O .... A Zt_i .... ^ ^^ — 2 . • . • 



Then writing A' = -^ («_i + «o) 



B' = 4 (^'"-2 + ^''^-i) 

 C = i (A%_3 + A%_2) &c. 



and a' = A?(_i, 5' = A^i/^j, c' = A^m+s &c. 

 there is the same relation as before between the successive letters, and the 

 formula is 



u^ = (A' + a'x) 



4}'2 _ 1 



4.r2 _ 1 ^o-^ — 9 4(;* — 25 o 



x(7D' + .'.) ^i-J %^« Tanr + •"'=• ^ 



Making a; = in this gives the well-known formula of bisection by sym- 

 metrical differences, yiz., 

 n _ A' --^ + 1-9 5C' _ 1.9.2 5___ 733, ^^^ 



2 



4 . 6 4 . 6 . 8 . 10 4 . 6 . 8 . 10 . 12 . 14 



-"^ 8^ +128^ 1024^ + 



FormulEe for quadratures by these symmetrical differences may be at 

 once obtained by integrating Newton's formulae between + limits,^ in 

 which case the terms involving odd powers of x disappear on integration, 

 leaving only the even differences in the formula symmetrical to an 

 ordinate, and the even mean-differences in the formula symmetrical to_ an 

 interval. The interval between the ordinates is assumed to be unity. 



Hence if there are n + 1 ordinates the limits will be ± -g ('i + l)> and 



