290 METHODS OF INTERPOLATION. 



ber of terms in eacli, we might find an nnlimited number of adjustment 

 formulas, but (14) and (15) will serve as specimens. Similar results can 

 also be arrived at by another method, which is very simple. We know 

 that in a series of tlie third or any lower order the fourth differences 

 are zero ; that is, any five consecutive terms are connected by the 

 relation — 



1^5 — 4 »4 + J<3 — 4l(.>+ Hi = 



and, consequently, we have — 



^':i = lV[^("2+«3+W4)-(?'l + «5)] • • . (10) 



This is identical with formula (14), which is thus shown to hold good and 

 to give exact results when applied to a series of the third order as well 

 as the second. It is therefore equally well adapted for graduating any 

 series, whether it has a point of intlexion or not. The same is true of 

 (15) and all other formulas derived from (13). 



When applied to an irregular series, such formulas can be modified so 

 as to give a<ljusted values which will approximate to the original ones 

 more or less closely, as may be desired. Take, for instance, fornnda (1(>). 

 If we add Jcu^ to both members of the equation next preceding, it will 

 stand — 



(10+A-)?ri=(4+A-)»3+4(?;2+»4)-("i+"5) 



and hence we have — 



This formula differs from (IG) in no respect except that the coefficient of 

 V:i within the bracket, and the denominator of the fraction without the 

 bracket, have both been increased by the same quantity I: Since Jc may 

 have any value whatever, we see that the weight of the middle term Uj 

 can be in(n'eased or diminished to any desired extent, the denominator 

 of the fraction without the bracket being increased or diminished by 

 the same amount. Thus if we desire that the weight of tij shall be 9 

 instead of 4, the formula will stand — 



W3=J^[9«3 + 4(W2 + W4) — (Wl + «5)1 . . . (17) 



In this way the value of each term in an adjusted series can be made to 

 <lepend on, and approximate to, that of the corresponding term in the 

 original series to any extent tluit may be required, and, of course, the 

 closer this ap[)roximatiou, the more nearly will the form of the new 

 series resemble that of the original one. 



When more than iive terms are to be included by an adjustment for- 

 mula, the relative weights of the terms can be varied by combining two 

 or more formulas together. For instance, (15) gives, if we drop the 

 accent from w'4, 



45»4 = 13(M3+"4+«5) + ^'=(«2 + «g) — 5(?/l+«7) 



