INTERPOLATION. 



171 



V = 



Subtracting now Y from V,, Y, from V,, and so 

 on, and putting 



F_ V y __ y 



~ *o _ v ' ' ' _ V */ 



- _ i ,--_ 1 . , sc. 



*! ^*0 'I - *! 



we find, 



V, = C+Z>(x 1 +x,+.r I )+&c. 

 Hence again we find 



V Y' :=ZXx - j }j. &c. 



r, r 



and putting- 



= 1\, we have V" =D+ &c. 



If, for the sake of ilistinct ideas,, we suppose the 

 function y to cor>ist of four terms or.ly, the operation 

 will terminate with this l.i.st equation. Taking now 

 the value of D, and returning to those of C, B, A, by 

 means of the expressions for 1 , V,,, and y, we find 



D = 



,+ 



B = Y Y '(x,+* )+ y o 

 A =y,Y ar .+ Y.'*s. 



These expressions being substituted in the value of y, 

 we get 



- 



. v- I'M*. -f *.+*)'' 



' I +(',*. +x I x c +x.*.>J- 



It is easy to see that the co-efficients of 1" , }'', V/, 

 nay be decomposed into simple factor*, and this 

 being done, whatever be the number of ordinate*, the 

 general term or expression for y may be put under this 

 form : 



(1) 



y=y.+ ! 



+ ) 



l) 



+ &C. 



The law of the series being evident, and, in this ex- 



/ . ' ' , Y, = , T t , 



*, x, x, jr, 'i x, 



V _J'4~~J'l IK- 

 r *^** 



V V \ V V V 



) - \ - r> ' V ' > '~ y JL- 



t^J ' r J *i = ^_^ i C. 



y y e jy y 



L 



This formula resolves the most general case of the pro- 

 blem. 



The general expression we have found for y admits 

 of another very elegant form, due to Lagrange, and 

 which has the advantage of being well adapted to lo- 



4 



garithmic calculation. If, in the expression for V ', Theory ef 

 we substitute the values of V, and V , we get 



*, X )(X, X )(X 3 *,) 



which, by putting , 5, y, for the co-efficients of y v y t , 

 ami j/,, takes the form 



In like manner V,, V,', &c. are found to be of the 

 form 



Again, if these results be put in the expression for Y ", 

 it will take the form 



and so on, , ft, y, ), &c. being put generally for cer- 

 tain co-efficients, independent of y {} , i/ lt y > & c - ar| d 

 consisting of various combinations ot'j . j,, &c. If, 

 now, these values be written tor Y,,, V ', &c. in the 

 general expression for.y, it will take the form 



where a, b, c, are co-efficients depending on the quan- 

 tities x,,, * x x_i, x, and c-r.tircly independ- 

 ent on u,, y t , &c. The values of a, b, c, &C. may be 

 easily determined for any particular ca-e, by following 

 the above process of substitution ; but they may also 

 be discovered, as it were, by inspection, if we recollect 

 that, when the indefinite abcissa x, by increasing gra- 

 dually from '/.rro (or 0) becomes x,, then the indefi- 

 nite ordinate y becomes y,. In this case, all the terms 

 alter the first ought to' vanish; so that a=l, 6=0, 

 c=0, . ... 1=0. But the quantities 6, c, d, &c. 

 on only vanish by reason of some common factor be- 

 coming =0. Now, we know that, in this particular 

 case, x x, =1) ; therefore x x c must be a factor of all 

 the co-efficients, except a. In like manner, it will ap- 

 pear that x x, must be a factor of all the co-efficients, 

 except b ; also, that x x 1 must be a factor of all, ex- 

 cept c, and so on. This putting A to denote some con- 

 stant quantity, we must have 



a=A x (x x,)(x JT.) .... (x x_i); 

 and supposing that x becomes x a , so that a = l, 



l=A X (x -*,) 

 and hence 



A = 



a =r 



(' *-)(*.*.) (*. * 

 In like manner, 



* a !-'!<*-'? -i*-H, to 



for none of the co-efficients can contain any power of 

 x higher than * 1. Substituting, therefore, these 

 value*, we get 



(2) 



(* *.)(*"-*.) (' *'-*)y 



"*"(* *.)(*!*) ' (*. **-l) y ' 

 4-&C. 



