234 ON THE MEAN RESULTS OF OBSERVATIONS. 



it will appear from what follows that the refinement of Cotes is 

 unnecessary; and, in the case under consideration, there are 

 practical reasons of another kind for adhering to the method of 

 equidistant observations, and which, therefore, deprive us of the 

 advantages of Gauss's method. 



3. Any periodical function U, of the variable x, may be 

 represented by the series 



U = A n + Ai sin (x + ai) + A 2 sin (2x + a 2 ) + A 3 sin (3a? + o s ) + &c., 



in which the first term, A , is the mean value of the ordinate 7", 

 and is expressed by the equation 



^o = 



This is the quantity whose value is sought in the present inves- 

 tigation. 



It is obvious that the values of U return again in the same 

 order and magnitude when * becomes x + 2ir ; so that if x = at, 



the period is represented by -^. If then 2* be divided into n 

 equal parts, so that the abscissae of the points of division are x f 



X + ~n' # + , &c., x + -, the sum of the corresponding 



ordinates will be 



-4 2 2 sin 



+ -4 3 S sin jsf x + ~\ + a 3 J + &c., 



in which denotes any one of the series of integer numbers, 

 from to n - 1 inclusive. The multiplier of A m , in the general 

 term of this series, is 



, K* 



- sin (mx 4 a m ] S cos + cos (mx + a m ) S sin ?^ 



But, when m is not a multiple of n, 



