292 . METHODS OF INTERPOLATION. 



These two conditions give the two values — 



so that the equation reduces to — 



and adding f iir, to both members, we obtain — 



Tlie same result can also be reached by deriving from formula (Ki) 

 any three special adjustment formulas comprising five, seven, and nine 

 consecutive terms respectively, and then combining them together in 

 the manner above indicated. There is evidently no limit to the number 

 of terms which might be included in formulas found by these methods. 

 With eleven terms, we have the following : 



in Avhich the weights are in arithmetical progression.* 



If we consider any seven consecutive terms in a series of the fifth 

 order, placing the sixth difference alone equal to zero, the equation thus 

 formed will give — 



u^ = ^^^[^r^{u:.+ }li+v,) — G{l(2+lk) + {Ui + n,)] . . (22) 

 This might be used as an adjustment formula, ])ossibly with good effect 

 in continuing the graduation of a series already ap[)roximately adjusted. 

 It will give exact results when applied to a series of the fifth or any 

 loAver order, and the weight of the middle term »4 can be increased or 

 diminished if desired. So, too, when the eighth difference is placed equal 

 to zero, we obtain the formula — 



W5 = ^i^[3G(?/4+«5 + «6) — -S(»3+«7) + S("2 + «8) — («l + «9)] • • (23) 



■which will give exact results if applied to a series of the sev(mth or any 

 lower order. 



The second method of adjustment can be api)lied to the logarithms of 

 a series of numbers instead of to the numbers directly. If, for instance, 

 the logarithms form a series of the third or any lower order, then for 

 any live consecutive terms formula (IG) gives — 



log «3=yV[^(log %+log 1*3 + log Ui) — (log »i+log «5)] 



= iW^S (»2W3«4)' — log («1«5)] 



and consequently — 



This relation will evidently hold good for any five consecutive terms 

 in a geometrical progression, because their logarithms are in arithmetical 

 progression 5 that is, they form a series of the lirst order. We can 

 easily see how any similar adjustment formula can be transformed at 



* For improved formulaa of tbia nature, see'Appendices I aud U. 



