THE BEST VALUE OF ^ 413 



ax 



(1) If tan 70 be denoted by M O , then tan 70 36' will be denoted 



and tto-e-fl + AVX 



= U + 0-6Aw - 0-12A 2 W + 0-056A 3 M - 0-0336A% 

 where AM,,, A 2 , A 3 ^, and A 4 w are the successive differences 

 running diagonally downwards from w . 

 Then tan 70 36' = 2-7475 + 0-6 x 0-1567 - 0-12 x 0-0168 

 + 0-056 x 0-0029 - 0-0336 x 0-0007 

 - 2-8396 



(2) If tan 77 be denoted by u , then tan 76 36' will be denoted 



KW-o-4 

 and w_ . 4 = (1 - 8)' 4 M 



= U - 0-48w - 0-128 2 w - 0-064S 3 M - 0-0416SX 

 /here Sw , S 2 M , 8 3 w , and 8 4 M are the successive differences 

 ming diagonally upwards from U Q . 



Then tan 76 36' = 4-3315 - 0-4 x 0-3207 - 0-12 x 0-0420 

 - 0-064 x 0-008 - 0-0416 x 0-0018 

 = 4-1976 



201. The Method of finding the Best Value of -^ from Tabular 

 r alues of x and y. 



Taylor's theorem states that if A = f(x] 

 Then ftx+h)- 



N W 



Hence f(x + h) -f(x) + hf(x] + .TM + -rto + 



If in a given set of tabular values of x and y, x and x + h repre- 

 sent two consecutive values of x, and j, x and u x+h represent the 

 two corresponding values of y. 



Then u x+h = u x + &u x = (1 + A)M B 



Also u x = /() 



Then u x+h =f(x+h) 



But 



dx h 



when /* is made infinitely small. 



