The Rev. H. Lloyd on the Mean Results of Observations. 63 



But, when m is not a multiple of n, 



2 cos = 0, 2 sin = ■ 



n n 



and, therefore, the preceding term vanishes. When m is a multiple of n, 



S cos = n, 2 sin — — = ■* 



n n ' 



and accordingly the term is reduced to 



n sin (7nz- + a,„). 



Hence, all the terms of the series vanish, excepting those in which m = kn k 

 being any number of the natural series, and there is 



^ 2(£/) = J„ + A„ sin {nx + a„) + A,, sin (2n.r + a,„) + &c. 



That is, the arithmetical mean of the n equidistant ordinates is equal to the sum 



of the terms of the original series of the order kn, whatever be the value of :r. 



The original series for U being always convergent, the derived series, 



which expresses the value of 1^{U), wiU be much more so; and, when the 



• These results are easily established. The roots of the equation ^ - 1 = 0, being 



comprised in the formula crx! _ j. / r ^\ ■ 2iV 



lormula cos -_ + ^/^ 1) sm _, the m« power of any one of these roots is 



'limir 2irmr 



cos -— + v'(- 1) sin -— - ; and the sum of the m«' powers of the roots is 



Sees + y/(- 1) vginf^_ 



" n 



Now, when m is not a multiple of n, this sum = 0, and therefore 



X cos = 0, 2 sin = ; 



™ n 



ther?forI' ''''" " " ' ""'""'' °' "' '""^ ^"" °' *<= """ ^^^^^ "^ *^ -°*^ = ". -^ 



2cosH^ = „, 2sinH^ = o. 

 n n 



This demonstration seems preferable to that derived from the general formute for the sum of 

 the smes and cosmes of arcs in arithmetical progression, which, in the latter of the two cases above 

 mentioned, lead to illusory results. 



