172 



INTERPOLATION. 



Theory of e ch term of which may be easily calculated by means 



JnterpoU- o f logarithms. 



We have now resolved the problem in its most ge- 

 "~ r ~"' neral form, nnd h.vc found two different expressions for 

 the indefinite ordinate y; it is, however, desirable, when 

 the nature of the case will admit, to have the ordmates, 

 or particular values ofy, at equal distances, and then the 

 differences ,,-* *.,-*,. **.. &c - re . a *** 

 In this case, lety.-y* y,y,, & the differences 

 of v v '/ *., y 4, &c. be taken in their order, and 

 then the difference of each of these differences, and so 

 on, as in the adjoining table, 

 where * is the difference be- 

 tween y , andjf, *,, the dif- 

 ference between y t and i/ ,, 

 and so on ; and, similarly, 

 is the difference between 

 o and ,, and y the diffe- 

 rence between ft a and ,, J 

 the difference between y and y,, &c. 

 A for the equal differences x, .r , 

 evident that 



co-efficient, but as a characteristic, to denote the differ- Theory of 

 ence between two consecutive values of the variable Intcrpola- 

 w. In like manner, *">" 



K, =A, 



, u 2 A 2 , 



y y* y* y, y*> &c - 



y. 



If we now put 

 x. Xr, &c. it is 



" "n i = A V n _ ,. 



If the function u vary by equal increments, the differ- 

 ences A , A ,, A u,, &c. will be all equal ; but if it 

 does not, these constitute a new series of quantities, 

 the differences of which may be expressed by the same- 

 notation, thus, 



A , A M = A A !< := A* 



A U 2 ,= A-A,= AX 



v * V *' V V -. &c. 



v._ r , Y,- T , y,_ h , *,- h , 



y'__^_ y' _ ^ 



2 A*' ' 2 A 1 ' 



a 



y y< 



* '~ 2l3A~ 3 ' YI = 2.3A 3 ' C> 



Y"- *" &c 



.-J*F- 



these values of F , Y[, Y^, V , &c. being substituted 

 in formula ( 1 ), it becomes 



,=,. + -^ o+ ^l(<^) a 



( 



1-2 A 1 



.)(*-*.)(*-*.) y . &c 

 1.2.3 A 3 



here the law of continuation is evident. 



If we suppose the abcissa x to begin at P, in the 

 above figure, the bottom of P Q, the first ordinate, 

 then x = o, x, = h, a, = 2 A, x, == 3 A, &c. ; and the 

 last expression may be put under this form, 



1 x I x I x \ 



y=y + y. -j- o + -i^~. y ^ ijto 



A !/ A _ ! = A-A u n _ ! = A'?< n _ i ; 



and here the figure placed over the A is to be undec- 

 stood, not as the index of a second power, but as indi- 

 cating a term of a second order of differences, originat- 

 ing from the variable function u. 



In like manner, by taking the differences between 

 the terms of the second order of differences, a third 

 order is formed, which are expressed thus, 



AX A*" = A 3 , 



Aa fl A _ i = A B _ i, 



and so on, to differences of a. fourth and higher orders. 

 By attending to the manner in which the quantities 



So, &C. 



have been formed from the series 



&c - 



the successive values of the function y, it will immedi- 

 ately appear that 



3 = A*y o , & c . 



so that, by employing the notation of differences to 

 the last general expression in both its forms, we shall 

 have 



(3) 



. ' r ~~*o A.. l_ \ X ~~ m)\ ~~ * ' &',i 



We have hitherto employed only the notation of the 

 Elements of Algebra ; but there is one particularly 

 well suited to this branch of analysis, which is in com- 

 mon use, when the quantities which enter into an in- 

 vestigation are the successive differences of the terms 

 of a series. 



To understand thq nature of this notation, let be 

 some variable quantity, and ,, a , tt 5 , a series of 

 consecutive values, which it acquires, either by vary- 

 ing itself, or else, in consequence of a change in the 

 values of some variable quantity on which it depends : 

 (For instance, u may be the abcissa of a curve, or else 

 the ordinal^)- Then, u t u is expressed by the sym- 

 bole A u, the Greek letter A being prefixed, not as a 



1.A 



1.2 



1.2.3 A 3 



also, supposing the abcissa to begin at the bottom of 

 the first ordinate, 



(4) 

 . 1 x 1 



