316 METHODS OF INTERPOLATION. 



equal to each other, and the operations of finding and verifying the 

 equation of the graduated series will become much more laborious. If 

 we do not know beforehand what form the function ought to have, the 

 most effectual means of increasing the accuracy of representation will 

 be to increase the number of constants equally with the number of 

 groups assumed. For instance, it is probable that a series of the sixth 

 order, obtained either by the first or the third method, will represent an 

 approximately adjusted series, such as (/) in Table II, more accurately 

 than any series of the fourth order, whether obtained with or without 

 the aid of the principle of least squares, can possibly do. 



The method of least squares can of course be used independently, for 

 the purpose of graduating an irregular series of numbers. But every 

 term will furnish one equation of condition, so that the number of equa- 

 tions will be as great as the whole number of terms in the series, and 

 if this number is large the amount of labor required to find and verify 

 the values of the constants becomes very considerable, while the method 

 cannot be expected to have any advantage over the method of interpo- 

 lation by groups, as regards the general accuracy of the result, except 

 in cases where the assumed function is capable of expressing the true 

 law of the natural phenomenon, or of approximating to it so closely that 

 the errors resulting from the difference in the form of the function will 

 be everywhere small enough to be neglected in comparison with the 

 errors of observation. Applied to an algebraic and entire function, the 

 general etfect of the method of least squares will be to increase a little 

 the accuracy of representation at the exti'emities of the scries, at the 

 cost of increased errors in the remaining portion. To illustrate this by 

 an example, let us compare two equations, taken of the second degree 

 for the sake of simplicity, each of them representing the first six terms 

 of series (A), the first equation being obtained by the method of groups 

 and the second by the method of least squares. In the three couvsecu- 

 tive grouj)s of two terms each the sums are — 



Si=.83107, S2=.79G80, S3=.8^473 



and since «i=2, formula (A) gives for the equation of the new series — 



w=.30717+.0()17075 X+.00512G2 x^ 

 If we assign to x the values — f, — f, — |, &c., in succession, the result- 

 ing values of u are the terms in the new series, as follows : 

 «i =.42494, «3=.397G0, «5=.4112G 



«2=.40G14, «4=.31)1)3(), «6=.'4334S 



"Wlien these are compared with the original values in series (/<), their 

 difterences or errors, taken without regard to sign, are found to be — 

 .001 7G, .00101, .00157 



.00177, .00100, .00158 



The sum of the squares of these errors is .0000132. 



!Next, we form six e<iuations of condition of.the second degree from 



