ON THE MEAN RESULTS OF OBSERVATIONS. 235 



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

 multiple of , 



2/W7T . 



S cos - = n, S sin -- = ;* 

 n n 



and accordingly the term is reduced to 



Hence, all the terms of the series vanish, excepting those in which 

 m = kn, k being any number of the natural series, and 



- S ( U] = Ao + A n sin (nx + a) + A 2n sin (2nx + a 3 ,) + &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 k)i> 

 whatever be the value of x. 



The original series for V being always convergent, the derived 



series, which expresses the value of - 2 ( U) t will be much more so ; 



and, when the number n is sufficiently great, we may neglect all 

 the terms after the first. Hence, approximately, 



The error of this result will be expressed by the second term 



* These results are easily established. The roots of the equation y" - 1 = 0, being 



comprised in the formula cos + V(- 1) sin, the m'* power of any one of these 

 n w 



roots is cos 1- V ( 1) sin ; and the sum of the m lh powers of the roots is 



limit ,. . 2imir 

 2 cos + v(-l) 2 sin . 



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



2tmir . limtr 



2 cos =0, 2 sin = 0, 



n n 



as above. When m is a multiple of n, the sum of the m"' powers of the roots = n, 

 and therefore 



2i;ir . 2wir 

 2 cos = ii, 2 sin = 0. 



