METHODS OF INTERPOLATION. 289 



aud let u' represent what S becomes when we take — 



n = l, a;=0 



then n' will be the middle term of the middle group Sj, and the lateral 

 groups Si and S3 will be similarly situated on each side of the middle 

 group and its middle term. We have then — 



This formula enables us to adjust the value of any term in an irregular 

 series, by taking it as the middle term with an arbitrary number of ad- 

 jiU'cut terms on each side of it, all together forming the middle group in 

 which the sum of the terms is S3 and their number is »2, Jmd taking two 

 other arbitrary groups, Si and S3, containing Mi terms each, and situated 

 one on each side of the middle term and equidistant from it. The dis- 

 tance from the middle point of the middle group to that of either lateral 

 group is ^1. The simplest case which can arise is where we take five 

 consecutive terms, ?<i, ii^, w-, W4, M5, and assume the three middle ones as 

 the middle group and the first oue and last one as the two lateral groups ; 

 then — 



112=0 J n\=l, rti=2 



and formula (13) gives, as the adjusted value of the middle term «3, 



= J^[4(«2+»3+»4)-(Wl+''5)]) 



When seven terms are taken, live in the middle groui> and t\vo in each 

 lateral one, so that the second and sixth terms belong to two groups 

 each, we have — 



n,=o, «i=2, «i=f 



and the formula is — 



w'4=J^[13So-5(Si+S3)] 



[(15 

 •?'e)-5(»i4-«7)] ) 



The accuracy of formulas (14) and (15) can easily be tested by trial with 

 any series of the second order, the adjusted value of the middle term 

 being in this case the same as its original value. A simple relation ex- 

 ists be ween the numerical coefiQcients of Wi, W2, &g. For example, in 

 formula (15) the coeflicient -1-13 belongs to three terniis, -f 8 to tw^o, 

 and — 5 to two, and we have — 



3x13 + 2x8 - 2x5 = 45 



and 45 is the denominator of the fraction outside the bracket. The 

 numerical coefficients within the bracket may therefore be regarded as 

 the wcifihtfi of the terms to which they belong, so that the weight of 

 each of the terms u^, Ui, and u^ is 13, that of ^12 and Vq is 8, and that of 

 Wi ami 11-: is — 5. 



By varviug the positions of the groups in formula (13), and the num- 

 19 s 71 



