330 KEPOET— 1880. 



indeterminately: ( h ——] u= — I udx-= 

 \ ax / h^ 



log (1 + A) 



= -'"(^+1-^-12-'^ + ^^^- >'' 



1 /^ rli 

 deterimnately : -v- / ? udx = A~^ u^ — \~'^u„ 



+ Y (^^^ ~ "«) 



12 



24 



- To C^"-- - ^^i) 



+ ^ (^hl,. - A\) - . . 



the first form, A~^ u^ — A"i i;,^ having the value already stated, namely, 



and ?{ taking the values u^ and «,, at the limit rh and qh. 



The coefficients here used are called the coefficients of quadrature ; 

 they are the coefficients of the expansion * 



= 1 + Via; + Y.^x' + Y^x^ + 



Y,= -^ 

 * 720 



v«= - 



60480 

 33953 •^ 

 3628800 



This is the leading theorem of simple quadratures, all the other 

 theorems being mere transformations or extensions of it. Before it 

 can be advantageously applied, some transformation is needed, because 

 both sets of differences run diagonally in the same direction, so that the 

 required differences cannot be got without the use of ordinates beyond 

 the limits of the integration. The simplest transformation is that by 

 ascendicg differences, which is easily obtained by expanding in terms of 



A . F 



F = instead of A = . But the most useful process is to ex- 



1 + A 1 - P ^ 



pand partly by ascending and partly by descending differences, whereby 

 we obtain symmetry, and use ordinates falling entirely within the limits 

 of integration. Observe that (1 + A) (1 — F) = 1. 



* See De Morgan Diff. and Int. Calc. p. 262 ; Lacroix, vol. iii. p. 182. Also 

 Woolhouse On Interpolation, S^c, (London, Layton, 1865), reprinted from the Assu- 

 ranee Magazine, vol, xi. 



