30 W. P. G. Bartlett on Interpolation in Physics and Chemistry. 



2, 2'^ &;c., indicate the algebraic sum of the s values of the 

 respective functions before which they are placed ; but before 

 taking the sum 2', the signs of all the numbers corresponding 

 to the cases in which the values of M are negative, and including 

 these values themselves, must be changed. Similarly before 

 taking ■2'", 2'"^ &c,, signs must be changed throughout for the 

 cases in which ^»i-, ^^^', &c., respectively are negative. So 

 ihaX 2'^t, ^''^''t"^ .^^rc^U^r are the absolute sums of these 

 quantities. The following equations 



(4) J-Ltnl^n ip —0, .2'['"]y[»]=:0 



are true for all values of n greater than m, and may therefore 

 be used as checks. Each of the conditions (4) breaks up (except 

 the case in which n=l) into two more convenient partial sums; 

 for, denoting the sum of all the values of a function correspond- 

 ing to positive values of .^"'i™ by -^^"'^(4-), and of those corres- 

 ponding to negative ones by ■2't'"^(— ), the equation 



2=0 is equivalent to 2'M(+)+2'W(-)=0, and 

 21ml— « « ^"»](-|.)_^«](_)=0; whence 



(5) ^-\+)=0, 21-l(-)=0, 



which may take the places of 2 and ^M in the form (4). There 

 might occur cases in which this principle of subdivision could 

 be carried on still farther. The advantage of using (5) instead 

 of (4) lies in the narrower limits within which it is necessary to 

 look for an error discovered by means of (5). 



The special forms of the various functions are written out in 

 (3) as far as will suffice for determining four terms in the value 

 of y, and computing y«^ so as to test the accuracy of the approxi- 

 mation and apply the checks (5) to it. An ins|iection of (3) will 

 show : 



1st, that the first term, ^, is simply the average value of y. 



2d, that to determine the second term it will be necessary to 

 compute y\ «,, ^t, and 31: 



Sd, for the third term, y", a^^ Jt^, i?j, ^-t'', and (t: 



4th, for the fourth term, y'", «,, Jt^, § ^, ^-/^ ^3, ^'f^ and 13: 

 and so on till the residual quantities, y^""^, are seen to be small 

 enough to be neglected. 



If more special forms are desired besides those written out 

 in (3), the law of their formation is obvious from an inspection 

 of those actually developed there. It is such that, in general, 

 if ^ and J^ be the wth letters in their respective alphabets, then 



It will be observed that no c 



