METHODS OF INTERPOLATION. 327 



Kow, to do the same thing by formula (A), we take — 



Si = 13, 82 = 21, 83=35, ni = n = l, 8 = % 



and so get — 



u = 2l + llic + 3^2 



which is the same as the other equation, except that x is reckoned from 

 the middle of the series. To obtain the interpolated term sought, we 

 take £i? = J, and it gives u = 27|, as before. 



This mode of procedure will sometimes be preferable to the ordinary 

 one, because the equation it gives will be arranged according to the 

 powers of x instead of according to the successive orders of differences 

 of the series. 



SECOND METHOD OF INTERPOLATION WITH AN ALGEBRAIC FUNCTION. 



This method, which gives an adjusted or mean value for the middle 

 one of a group of any odd number of terms, by assigning certain "local 

 weights'' to the several terms of the group, was at first regarded by the 

 writer as chiefly useful in making a rough adjustment of a given series, 

 preparatory to the application of the first or third method. It appears 

 now, however, that very little can be gained by such preparatory adjust- 

 ment; the errors obviated by it being probably smaller than other errors, 

 which are almost necessarily incurred by employing the first and third 

 methods when the true law of the series is unknown. But the second 

 method, in its improved forms, is quite worthy to hold a place of its own 

 as an independent system of adjustment. In the previous memoir, (pp. 

 334 and 335,) some doubt was expressed as to which system of local 

 weights is the best one ; but it now seems clear, from the following consid- 

 erations, that the preference should be given to formulas (69), (71), »&c., 

 which render the probable value of the fourth differences of the adjusted 

 series a minimum. On the assumption that the adjusted series ought 

 to be continuous, so that any five consecutive terms in it can be regarded 

 as agreeing very nearly with a curve of a degree not higher than the 

 third, it is easily shown that a minimum value of the probable fourth 

 difference implies a minimum of probable error in the corresponding 

 term of the series. Let «fi, th, &c., be any consecutive terms in a series 

 of the third or any lower order; then we hnve — 



J4 = 6U3 — 4 {ti2 + ^4) + ("i + W5) = 



But if each term is subject 1:0 a small error of accidental nature, whose 

 Ijrobable value is e, then the jirobable value of J4, taken without regard 

 to sign, becomes — 



(/J4) = e \/6' + 2 (4'^ + 1) = £ ^/70 = 8.3666 s 

 and consequently — 



£ = .11952 (4) . (84) 



We thus see that the probable error of a term bears a fixed ratio to the 



