METHODS OF INTERPOLATION. 317 



the first six terms iu series (/<), and find tliat by the method of least 

 squares the equation of the new series is — 



« =.39710+. 0015743 .r+. 0051468 x" 



This gives for the terms iu the new series — 



«i=.42533, «3=.397G0, '?^5=.41104 



«3=.40G32, «4=.39918, W6=.43321 



the errors are — 



.00137, .00101, .00179 



.00195, .00112, .00131 



and the sum of the squares of the errors is .0000129, which is a mini- 

 mum. Comparing these results with the ones obtained by the method 

 of groups, we see that nothing has really been gained in accuracy by 

 employing the method of least squares, since the maximum error has 

 been increased by it from .00177 to 00195. Besides, the method of 

 groups has a great advautage in the simplicity and brevity of the cal- 

 culations required.* 



The sum S of the terms in any group can be expressed in still another 

 form b}^ means of a series. When /(.r± in) is expanded according to 

 the j)owers of hi^ it becomes — 



f{x:^ i n) =f{x) ^f'{x) 0^ + lf"{x) Qy ± ^^'"{x) (^) ' 



+2:^-^"(^"Kl)'=^ ^^^"- 



where /'(.r), /"(a-), &c., are the successive differential coefficients of /(j?). 

 Consequently we have — 



^=f{x-\-^n)-f{x-hn) 



y/"H^-)Q)%&c.] 



.(45) 



^2.3.4.5.6 



This series will terminate \f f{x) is algebraic and entire. To illustrate 

 its application, let us assume — 



f'{,x)=A^^x+Cx' 

 then the other derivatives are — 



/"(x-)=B+2C.» 

 f"'{x)=2G 

 while f'^{x), /""{x), &c,, are zero. We have accordingly — 



=?i[A+Bx+C(.i'2+ j-L n-)J 



* There is still another method of iuterpolation, devise*] by Cauchy, which can be 

 used iu cases of this kind. It is, however, more laljorions than the method hero pro- 

 posed, and trials which have been made indicate! that it does not secure any greater 

 accuracy. For some account of it, see the American Journal of Science for July, 1862, 

 and LionviUeg Journal, vol, 18, page 299. 



