INTERPOLATION 411 



allowing a certain amount of latitude for this, let the values of 

 A 4 p be neglected. 



(1) To find the value of p when 6 - 106. 



If = 105 be denoted by U Q , then = 106 will be denoted by M 0>2 



and M . 2 = (1 + A) - 2 ^ 



= M + 0-2Aw - 0-08A 2 M + 0-048A 3 M 



yhere AM O , A 2 ^, and A 3 M are the successive differences running 

 diagonally downwards from U Q = 105. 



Then W . 2 = 17-53 + 0-2 x 3-27 - 0-08 x 0-47 + 0-048 x 0-08 

 = 18-15 



Or if = 110 be denoted by UQ, then = 106 will be denoted 

 by w_ . 8 



and w_ . 8 =(l-S)X 



= M - 0-88w - 0-08S 2 M - 0-0328 3 M 



where SM O , 8 2 M , and 8 3 M are the successive differences running 

 diagonally upwards from U Q = 110. 



Then M_ . 8 = 20-80 - 0-8 x 3-27 - 0-08 x 0-44 - 0-032 x 0-05 

 = 18-15 



Hence when = 106, p = 18-15. 



(2) To find the value of p when = 92. 



If = 90 be denoted by M O , then = 92 will be denoted by M . 4 



and M . 4 = (1 + A)' 4 M 



= M + 0-4Aw - 0-12A 2 w + 0-064A 3 M Q 



where Aw , A* , and A 3 w are the successive differences running 

 diagonally downwards from w = 90. 



Then w . 4 = 10-16 + 0-4 x 2-10 - 0-12 x 0-34 + 0-064 x 0-05 

 = 10-96 



Or if = 95 be denoted by u , then = 92 will be denoted 

 by tt-o-e 



and M_ . e =(l-S)-X 



= M O - 0-6M - 0-128X, - 0-0568X 



where 8w , 8 2 M an( ^ ^ 3w o are the successive differences running 

 diagonally upwards from M O = 95. 



Then M_ . e = 12-26 - 0-6 x2-10 - 0-12 x 0-32 - 0-056 x 0-06 



= 10-96 

 Hence when = 92, p = 10-96. 



