420 PRACTICAL MATHEMATICS 



The above table gives some of the successive differences which 

 have been obtained by working backwards from the seventh 

 difference column, assuming that AX *=* 0. 



* 



dx 



- 1 ( A% + SwJ - \ (AX - SX) + i (AX + SX) - g (AX - SX) 



+ ^ (AX + SX) - ^ ( A6w * - S6w *) + H (AX + SX) 

 But AX + 8X = 



AX ~ SX = 



& 5 u x + S 5 w x = 2b + a 



A% x - 8X = 4& + 2a = 2(26 + a) 



5b 



AX + ^ 3w x = %d+c+--+ a 

 LI 



A 2 w x - SX = 2d + c + 5 



SI 



Hence ( AX + S 3 wJ - ( AX ~ X) = 26 + a 



and AX + SX - (AX + &X) ~ (AX - 2 %) 

 also A% x - 8X = 2 (AX + &X) - 2(A 2 w a; - X) 



Therefore /i -5^ 

 a>r 



- 1 (Aw, + Sw z ) - \ (AX - 8X) + \ (AX + SX) 



- i {(AX + S 3 wJ - (AX - SX 



To 



) - (AX - 



(Aw, + K) - (AX - X) + (AX + X) 



) - - - (1) 

 (2) A 2 - i (AX + SX) - (AX - SX) + (AX + SX) 



- A (A 5u x - SX) + (AX 



