METHODS OF INTERPOLATION. 325 



With decimal weights it becomes — 



W4=.4o998?(4+.29281(»3+W5) + .01752(«.+i/6)-.04032(»i+«,) 



In a simikir manner we might proceed to And formulas including more 

 than seven terms. With nine terms we should assume acurve of the tenth 

 degree, with the three conditions that its first, second, and third dif- 

 ferential coefficients should all become zero at the positions of the two 

 nearest zero weights. 



This method of determining the weights may seem to be theoretically 

 better than the previous one, but the labor required in obtaining the 

 formulas is very considerably increased, especially when nine or more 

 terms are to be included by them, and the practical advantages of the 

 method, if it has any, must be small.* According to the theory of 

 probability of errors, if we let e denote the probable error of each single 

 term in a given series, then the probable error of a term adjusted by 

 the above formula will be — 



eo=e V.459982-f2(.2928P4-.017522-f .040322] =.62204 ^ 



But if the adjustment were made by formula (54), the probable error 

 would be only — 



eo=e 7.41470^+ 2{.29480-'+. 05494-'+. 05718^) =.59874 £ 



which indicates that (54) is slightly superior in the accuracy of its 

 results. This, however, is not conclusive as regards smoothness of ad- 

 justment. If we imagine two series, such that the probable error of a 

 single term is smaller in the first one than in the second, it is still pos- 

 sible that the second may be the more perfectly graduated of the two, 

 since its errors may follow a continuous sequence or curve, while the 

 errors of the first may be arranged irregularly or fortuitously, so as to 

 follow a broken line. The comparative regularity of the graduation of 

 two series obtained by using different adjustment formulas will be best 

 ascertained by comparing their corresponding orders of differences. 



The fourth difference is most convenient for this purpose, and may be 

 obtained directly for any five consecutive terms by means of the for- 

 mula — 



Ai = G ^3 — 4(«24-«4)+(Wi+?<5) 



Having thus computed all the fourth differences for each of the two 

 series, we can add them together in each case without regard to sign, 

 and the series which gives the smaller sum may be regarded as the 

 better graduated of the two. This becomes evident when we consider 

 that a curve of the third degree, since it admits a point of inflexion, 

 may be taken to represent ai)proximately a limited portion of any 

 regular curve ; and as all !he formulas of the second method of adjust- 

 ment give accurate results for a series of the third or any lower order, 

 their use tends to bring the adjusted series into such a form that any 



* Subsequent trials Lave sliowu that it has none. 



