METHODS OF INTERPOLATION. 327 



to be adJHsted is not a very irregular one. We can proceed in a simi- 

 lar way to obtain formulas for adjusting the middle term in any group 

 of tive, seven, nine, or more terms, as follows :* 



«3=3V[1^ "3+12(W2+«4)-3(?/l+H,)] 



In all these cases, the weights form a series of the second order. The 

 probable errors are less than those given by other similar formulas ; for 

 instance, the probable error of the adjusted value of u^ is only — 



— ri77:\^c- 



But it has been found on trial that, as regards smoothness of adjust- 

 ment, these formulas are decidedly inferior to (53), (54), &c., or even to 

 (17), (19), &c. This is owing to the great want of continuity between 

 the weights of the formula and the zero weights. If we apjjly Cauchy's 

 method to the same series of terms as above, w^e get — 



^'3 = To[4(W2+«3+«0 — (^'l + «5)J 



«*4=3V[11(«3 + «'4 + W:.) + 4(«2 + Wg)-3(«i + «7)] 

 %=2t[^("3+W4+«5+"g + W7) — (Wl + «2 + «8 + «9)] 

 ^'6=2^T['il(«3+W4+ +«9)— 14(Mi + «2 + «10 + '»<n)] 



All these, except the second, are special cases under our formula (13). 

 The first one is the same as (14). 



ADDITIONAL FORMULAS UNDER THE FIRST METHOD. 



The simplest case of all has been omitted ; it is that in which the 

 graduated series is of the first order, so that the expression for the sum 

 of any n terms in a group is — 



Assuming any two groups composed of Ui and ih terms respectively, 

 with the origin of coordinates midway between the middle points of the 

 groups, and denoting by a the distance from the origin to either of these 

 points, we have for the values of the constants — 





(G- 



* lu like manner, it can be shown that formulas (48), (40), and (50) are in accord- 

 ance with the principle of least squares. 



