350 METHODS OF INTERPOLATION. 



A great drawback to the utility of this whole method of interpolation 

 is that the equation of relation will sometimes be found to have one or 

 more negative roots, in which case the method fails, although it may 

 succeed if we assume a different set of groups. It is evident that if we 

 have, for instance, (3^ negative, then 1^ will be either negative or imagin- 

 ary; and when values diifering from each other by unity are successively 

 assigned to x in Px, the resulting values will be either alternately posi- 

 tive and negative, or some of them will be imaginary, and in either case 

 the series becomes useless for purposes of interpolation. 



It should also be observed that in the process we have followed, the 

 roots of the equation of relation are all supposed to be unequal. If any 

 of them were equal, it would involve a change in the form of the func- 

 tion. For instance, if there were m real roots equal to P^, they would 

 together correspond to a term in the function of the form — 



1) {1 -[- ai X + az x'^ ■{- + am-ia;™-^),^^ 



while, if there were m equal pairs of imaginary roots, they would be 

 represented in the function by a term of the form 



{l + aiX+a2X^+ . . - . . - +am~i^'^~^) Ic sin {x 0) -{■ d ao^ {x e)]^'' 

 But the case of equal roots is a special one, which can hardly be expected 

 to occur when the equation of relation has been constructed from an 

 irregular series of numbers derived from observation ; so that, for X3rac- 

 tical purposes, it is not necessary to consider this branch of the subject 

 any further. 



We see from the above that an algebraic and entire polynomial is a 

 special case under the class of recurrent functions, the case, namely, in 

 which all the roots of the equation of relation are equal to unity, so as 

 to make /? = 1, and consequently I3x = 1. That any series of algebraic 

 form is recurrent, is evident when we consider, for example, a series of 

 the third or some lower order. To say that any five consecutive terms 

 in it are characterized by the property — 



J4 = 6 % — 4 («3 4- iti) 4- (Ml 4- U5) = 

 is merely saying that any term tt^ can be obtained from the four terms 

 next preceding it by multiplying them severally by the scale of relation — 



-Ai=-1, -A2 = 4, -As = -6, -A4 = 4 



and adding the products together. Upon this recurrent property 

 depends the construction of all the adjustment-formulas of our second 

 method. They, however, are not the only ones which can exist under 

 the general form (88), for we can assign any real values we please to the 

 weights lo, h, h, &c., and the formula will still apply to some transcen- 

 dental series. For instance, we have seen that the formula — 



^3 = /t^tg I ^ ^'3 + 4 («2 + th) — (wi 4- 115) I 



in which h is any arbitrary number, will give exact results when applied 

 to a series of the third or any lower order; tha,t is, to any equidistant 

 values of the function — 



u = A ^B X -\- G x~ + J) x> 



