358 REPORT — 1880. 



- 21^^ (^^^3 + A«U.) + jJ^f^, (ASU, + ASU3) - . . . . 



A^ ^24 ^ 6760 •' ^ 1890 . 29 ^ "^ 56700 . 21* 

 ^,y 1295803 ^loy^ _ 15183675231 ^,, 



^ ^ 1871100 .216 ' 7662154500 .220 ^ ^ ^ 



__289_ 

 56700 . 26 ■ 



Most of Legendre's tables of definite integrals were computed by the 

 second or third of these formulse. His great table of elliptic functions 

 was calculated by the second. 

 For differential coefficients 



:A8U3+ 



= ^Vi + ^, A3V, + ^^ A'3V3 - ^- A7V, 



— , A3V, + -A_ A5V, ^ 



3 . 23 ' ^ 6 . 27 ^ 7 . 2'o 



35 



9.215 

 If TO. = in the N formula, it becomes 



1 ,Tr . Tr^ 1 . . ,Tr . . ,x. ^ . 3 



+ ;r^. A9V5- 



^* = ^ (Vi + V) - 1^ (A^V^ + A^VO +j~^ (A^Va + A^V^) 



- in (^'"^4 + A6V3) +|| (A«V5 + A«V,) - , 



a well-known formula of bisection. 



Some obvious transformations will enable corresponding formulte to 



be obtained for A" / ""dx" where in and n are different.* 



It is to be observed that the formula for integration above given is 



not ludx, but - A jtulx, which is a different thing. If / mZa; itself be 



required, it must be obtained by summation. 



A very compendious method of interpolating tables by means of 

 bisection is derivable from the ordinary formula of bisection, by affecting 



it with A, Thus, stopping at the first term, namely, u ■=■ «(Vi + V) 



2 



A« = |(AVi+AV) 

 = AVi + ^A2Vi 



:=AV-iA2V. 



* See Woolhouse on < Interpolation, Summation, &c.' in the Assurance Magazine, 

 vol, xi. p. 301 et seq, for some interesting transformations of these theorems. 



