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



certain definite abscissa3.* But in the case of periodical functions, 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 ad- 

 hering 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 = Ao+ Ai sin {x + ai) + A2 sin {2x + o^) + ^3 sin (3a; + 03) + &;c., 



in which the first term, Ao, is the mean value of the ordinate U, and is ex- 

 pressed by the equation 



1 f+" 

 -^0- K-\ Udx. 



This is the quantity whose value is sought in the present investigation. 



It is obvious that the values of U return agam in the same order and 

 magnitude when x becomes .t -1- 27r; so that if .5; = at, the period is represented 



by — If then 27r be divided into n equal parts, so that the abscissae of the 

 •' a 



Sir i-n . 2(n - l)7r „ , 



points of division are ,r, .r -f — , a- -i- — , &c., .r + -^ , the sum ot the 



i^ n n n 



corresponding ordinates wiU be 



•2 { U) = nAo + A,^ smU+-^ + aA + A,^ sin|2 h: + -^j + °2 j 



+ JlsS sin I 3 f a; -f — j -1- 03 l + &c. 



in which i denotes any one of the series of integer numbers, from to ?i - 1 

 inclusive. The multiplier of J„, in the general term of this series, is 



1, sin I ml s + — j + a„ > 

 = sm (m.r + a„) 2 cos h cos (?7ix + o„) S sm 



71 ^ ' n 



Commentationes Societatis Regke Scimtiarum Gottingensis, torn. iii. 



