METHODS OF INTEEPOLATION. 313 



pie seems to indicate that so far as has yet been ascertained, tlie most 

 advantageous mode of grouping is to make the groups consecutive and 

 composed of an equal number of terms each; a system which has, 

 besides, the merit of greater simplicity.* 

 The algebraic and entire function — 



2/=A+Bx-+C^+&c. 



is of course not the only one which it is possible to employ for the purpose 

 of graduating a given irregular series. If we take any other continuous 

 function — 



2/=^(A,B,C, T,x) 



then, as before, the integral — 



S 



HA, B, C, T,x)>lv 



will express the sum S of the terms in any group in the graduated series 

 by means of the number w of terms which that group contains, the 

 abscissa x of its middle point, and the constants A, B, C, . . . T. 

 By assuming in the given series as many grou])S as there are constants, 

 and giving to S, «, and x their numerical values taken from these several 

 groups, we shall have as many equations of condition as there are con- 

 stants to be determined; and if we can perform the operations necessary 

 for finding the numerical values of the constants from these equations, 

 then the equation of the graduated series can be easily formed, and the 

 series itself can be constructed therefrom. This series will not have any 

 one of its orders of differences constant, but it will be a graduated 

 series nevertbeless, and the arithmetical means of the terms in the cor- 

 responding groups in it will be severally equal to those in the original 

 series. It will, no doubt, sometimes be possible to lind in this way a 

 transcendental equation which will express a given series more advan- 

 tageously than an algebraic equation could do. 



We may here notice a peculiarity of the circular function — 



2/=A + Bsin(%^')+Ccos(^^^)+Dsin2(=^) 



-fEcos2^?|^^VFsin3(^'''VGcos:^(^-''V&c. 



in which N denotes the number of terms in the circular period, or the 

 length of the period measured on the axis of X, so that if the values x', 

 .t'+N", A'-fliN^, &c., are successively assigned to x, the value of y will 

 remain unchanged. The arithmetical mean of any n terms taken in a 

 group, and also the mean value of the ordinate within any interval n, 

 will be — 



,^ s 1 r^+in 



M=-=- / ydj 



*Tliis may bts a too hasty conclusion. Other trials have since shown that (4i)), (41), 

 and (4-i) do suiuetimes, and perhaps generally, give the best results. 



