METHODS OF INTERPOLATION. 203 



•onee in this way. The weights of the several terms become their expo- 

 nents, the terms with positive weights become lactors in the numerator 

 of a fraction, wliile those with negative weights are factors in the denom- 

 inator, and tlie fraction without the bracket becomes the exponent of 

 the whole. Thus (22) is transformed into — 



''''\ u^? ; 



which expresses a relation existing between any seven consecutive terms 

 in a series whose logarithms form a series of the fifth or any lower order. 



In all formulas under the second method, the weights of the several 

 terms, depending on the position of each one with reference to the mid- 

 dle term whose adjusted value is sought, may be called local weights, to 

 distinguish them from the intrinsic weight which any term may have 

 by virtue of the relative goodness of the observations taken to deter- 

 mine its value. We may regard the total weight of a term as com- 

 pounded of these two elements, so that if, for instance, the local weights 

 of tive consecutive terms are taken as in formula (1(5), and if Ave wish 

 also to take the intrinsic weights Cj, c^, c^i &t'', of the terms into account, 

 the adjusted value of M3 will then be — 



_ Hc.no+c^ih+CiUi) — {c'lVi + Crjis) ,^,. 



4(e2+C3+C4)-(Ci + C5) '-'[-) 



We know that this fornnda gives exact results when the series Wi, ^2, &c., 

 is of the third or any lower order, and the intrinsic weights Ci, C2, &c., 

 are all equal, and we may naturally expect that the results will be 

 approximately correct when the series 2*1, «o, &c., approximates to regu- 

 larity, and the intrinsic weights of the terms do not differ very much 

 from one another ; so that in such cases something will be gained in 

 accuracy by taking the intrinsic weights into account. 



By the use of formulas such as (10), (17), (10), or (20), we can grad- 

 uate approximately all the terms in a series except the first two and 

 last two. Tliese also can be reached by means of the general formula 

 (2). Let us take six consecutive terms in three groui)s, so as to have — 



wi=;>, 7/2=2, n'j=l, «i=f, «3=|j n — 1 



Then for the first term we have — 



and the formula reduces to — 



<<,=.r(5Si_,r-,S2+4S,) ^ 



For the second term we have — 



and consequently — 



«;=j_(14Si+4S2-5S3) ; 



(25) 



(20) 



