170 



INTERPOLATION. 



Theory of whole, a series of the same nature as before, but con- 



Intcrpoli- 8 i s ting of more terms. Thus, in the series 1, 3, 6, 10, 



tion. f x i ] J ( x i g) 



" -/"' &c. of which the general term is - 



we 



may interpose three terms between every adjoining 



two, by making x = $ 



also = 1^, If, Ij, &c. 



The terms of the series and their indices will then be 



, 



\, 1, 1, &c. 

 tf, 3, 7 V, &c. 



When new terms are thus interposed between the 

 terms of a given series, they are said to be interpolated. 



If the law of formation, or general term of a series 

 be given, there is no difficulty in interpolating any 

 term, of which the index is given : however, in the ap- 

 plication of this theory, it for the most part happens, 

 that the general term is unknown, and only some par- 

 ticular values of it are given, by wjiich it is to be found, 

 or at least some approximation to it. 



As the general term and its index are perfectly ana. 

 logous to the ordinate and abcissa of a curve, the ana- 

 lytical problem to be resolved is manifestly the same 

 as this geometrical one : Having given a determinate 

 number of ordinales and the corresponding abcissts, to 

 Jind the equation of the curve ; or to describe a curve 

 through a certain number of given points. It was under 

 this form that Newton first proposed and resolved the 

 problem. 



Let P Q, P' Q', P" Q", P'" Q'", P lv Q", P Q T , &c. 

 (in the annexed Figure,) be the given ordinates, or 



particular values of the general ti'rm y, which may 

 stand either at equal or unequal distances from each 

 other, the positive values P Q, P' Q', &c. being sup- 

 posed to lie on one side of the axis AB, and the nega- 

 tive values P' v Q", P v Q v , &c. on the other; and let 

 AP, AP', &c. be the given corresponding abcissae, or 

 particular values of the variable x. Also let PQ=y be 

 any indefinite ordinate, or term to be interpolated, and 

 AP=,r the corresponding abscissa, or variable index of 

 that term, and, lastly, let CDE be the curve, passing 

 through the tops of all the ordinates. The thing to be 

 done is, to express y in terms of x, or to find the nature 

 of the curve. 



If there be no datum, in addition to the condition, 

 that the curve pass through the given points Q, Q', 

 Q, &c. the problem is manifestly indeterminate, be- 

 cause, by varying the species, any number of curves 

 may be conceived to pass through the given points. 

 I lowever, in the applications, the ordinates are in ge- 

 neral near each other, the distances between them not 

 very unequal, and the curve is known to have no con- 

 siderable inflection between adjacent ordinates. Under 

 these circumstances, we may assume that between 

 1" Q', and P", Q", any two adjacent given ordinates, 



the indefinite ordinate y=VQ may be expressed by a 

 series composed of the powers of the abcissa P' P, thus, 



y=a + b X (P'P) + c X (>P)' + </X v (P' P) 3 + &C. 



the quantities a, b, c, &c. being supposed constant ; 

 and as the series will converge, because P' P is small, 

 a few terms at the beginning will be a near enough 

 approximation to its value : Or, since AP=r.r is equal 

 to AP'+ P'P, therefore, P'P= j AP': Substituting 

 now this last quantity instead of P' P, and remarking 

 that AP' is a given quantity, by hypothesis, we shall 

 have 



y = A + Bx + Cx ! + D & + &c. 

 A, B, C, &c. being put for constant quantities : Curves 

 having their equation of this form are called parabolic, 

 from its analogy with the equation of the common pa- 

 rabola. 



This expression for the ordinate^ in the geometrical 

 problem, is, considered analytically, the general term of 

 the series to be interpolated, or at least, an approxima- 

 tion to it, which will be sufficiently near in tne ordi- 

 nary cases to which interpolation is applied. 



The form of the expression for the ordinate or gene- 

 ral term being settled, we are now to find the constant 

 co-efficients A, B, C, D, &c. 



Let the given terms, or values of y, which in the 

 curve are represented by the ordinates P Q, P' Q', 

 P" Q", &c. : be denoted by the symbols 



and their indices, or the corresponding values of .r, 

 which in the curve are the abcissae AP, AP', AP", 

 &c. by 



since the general equation 



y = A + B x + Cx' + D .r* + , Sec. 



holds true for every pair of corresponding values of if 

 and x we have, by substituting in it y and x a , then 

 y l and x,, &c. 



y = A + B x, + C x* + D xl + &e. 

 y t =:A-\-Bx, -\- C xl -f- Do-J -j- &c- 

 y*=A + Bx, + Cxi + I>*1+ &c. 



y , = A + Bx i +c*j +DX; +&c. 



The number of these equations must be equal to the 

 number of constant, but indeterminate co-efficients 

 A, B, C, D, &c. and as only the simple powers of these 

 enter the equations, they may be all determined as 

 follows. 



By subtracting successively the first equation from 

 the second, then the second from the third, and so on ; 

 and afterwards dividing the respective results by 

 x t , * 2 *,, *, x t , &c. we obtain 



x, 



)+ & c . 



+ &c. 



Putting now, in order to abridge, 



we have these equations 



