9^ Prof, Encke on Intetpolation. 



nearest term in order to form x — a, and then on the other side 

 i^x—h), and continue thus taking differences alternately on one 

 and the other side, the order of the factors is then entii'ely to 

 be reversed for the use of the formula (VI). 



These latter formulae have this important advantage, that 

 in them no regard need be had to signs, if the following rule 

 is attended to : — that all the difference-quantities must be so cor- 

 rected as to be brought nearer to the above-mentioned line, 

 above and below which they alternately lie, or that the correc- 

 tion must approximate every one to the difference-quantity on 

 the opposite side of the line. In order to perceive the reason of 

 this, let the two cases in which the factor of correction is of 

 the form .r — «„, and in which it is of the form x—b, be distin- 

 guished. In the former the correction is always 

 ■\-(x—a)\a a ,.,.ab...b ,il 



For brevity let us assume the particular case of?? = 1, and 

 making the proper arrangement of the quantities it will be 

 found that the rule requires, that [a, a 6] 4- (•^ — «i) \^cc,ab 6,] 

 should always be between [a b i,] and [a, a &]. But we have 



by what was shown above Va.abb,'] = -^ — , ^ ; conse- 



•' 1- . ij 0,-0, 



quently the expression (C) becomes 



= a,ab + -rzr~ j [^ ^ ^i] ~ C^j ^^2 \ 



and the factor ^_"^ ■ is in all cases, by the notation adopted, 



positive and less than 1. As the quantity [a b i,] may be thus 

 expressed [a, a & ] + [[« b 5i] — [a, a i]], it is clear that the 

 correction tends, agreeably to the rule, to approximate the 

 quantity [c, a b] to [_abb,'], with the exception of the single case 

 in which a former correction had changed the sign of the 

 quantity [a, a b 6,]. In this single case of exception there 

 will be a further removal from that quantity. But with some 

 little attention, especially in performing several interpolations, 

 it will not be possible to make a mistake on this head. 



The same will take place in the second case, in which the 

 factor of correction is of the form x—b . The correction is 



n 



(x—b ) [a . . a ... a b ... b 



\ n' L n-\-\ n n 



to be applied to [a^^ ... ab ... i^J, which according to the rule 

 it is to approximate to the quantity 



[a . . a ... a b ... b ,1 

 The two expressions become in this case P 



