332 , EEPOET— 1880. 



then, neglecting all furtlier terms, as a second approximation 



For a third approximation 



■ iX^'l.,- 1) { A2^o + i Oi - 2) A3 ^0 j = C2 



may be calculated : tliis gives 



The process is very cumbrons when carried beyond the first correction of 

 the proportional part. But it has one very marked advantage, namely, 

 that, being a tentative process, any en'or in one step is more or less com- 

 pletely corrected at the next step, and the practical effect of accidental 

 erroi* is thns to make the approximation less rapid, instead of absolutely 

 vitiating the result. It might even happen that an error of calculation, 

 by being nearer {he required answer, might give a more rapid approxi- 

 mation. 



The rapidity of approximation depends, firstly, upon the degree of 

 convergence ; and, secondly, upon the first approximation being suSiciently 

 near the required result. This is exactly parallel to what takes place in 

 the numerical solution of equations, and there is, here as there, the same 

 difficulty, presenting itself where any given approximation is nearly half- 

 way between two solutions, and the successive results oscillate between 

 the two, instead of convergingr to either.* 



When convergence is assured, this tentative method is probably the 

 best, and is at any rate the safest. An extension of Hutton's rule for 

 extracting roots t might possibly be found of use. But the criterion 

 of convergence in this process has not been satisfactorily determined. 

 There is, however, reason for believing that the convergence is not so 

 good as in the direct tentative process given above. 



Tliis process is applicable, not only to the common formula of inter- 

 polation by descending differences, but to all formulae which can be 

 arranged by ascending powers or factorials of n, the index of interpolation 

 For in any such form it still remains as the approximate solution with 

 respect to x of an equation of the form 



y = a^ X + a^x^ + as x^ + . . . . . . 



where f«™ is either a power or a factorial of x. 



Section 3. — Equidistant Ordinate^, not differenced. 

 In genei'al, writing 



Ih = Co "o + Ci «i + C„ tl-n 



where u^u^ are certain given values of u corresponding to given 



values of x, namely, a-g, x^ . . . . x,„ and then assuming a form of u in 

 terms of x which will allow the coefficient c to be so determined that 

 X = a3,..shall make c,, = 1 (when x,. is one of the given values) and all the 

 other c's vanish, a formula of interpolation is obtained which can be con- 

 verted into a formula of quadrature by integrating with regard to x from 



* See Horner, in Leyhonrns Ecpomtory, No. 19, p. G3, and J. K. Young, Tliem-y 

 and Solution of the Higher JSquations, second edition (1843) note, pp. i74-6. 



t For which seethe London, Edin.and Biihlin Philos. Mug. vol. xs. (I860) p. 440, 

 and the Philos. Trans, for 1802, vol. clii. pp. 429-431. 



