181 



or meteorological element, for any day, may be had approxi- 

 mately, by taking the arithmetical mean of any number of 

 observed values obtained at equal intervals throughout the 

 twenty-four hours; the degree of approximation, of course, 

 increasing with the number. It is important to ascertain the 

 law which governs this approximation. 



Any periodical function, u, of the variable y, being repre- 

 sented by the formula 



M = «o + «i sin (v + a\) + a.2 sin {1v + a^ + &c., 

 in which a^ is the true mean, or 



^o^-^A «^^' 



if Ml, U2i Us, 8ic.,u„, denote the values of the function u, cor- 

 responding to those of the variable 



27r 47r „ 2(w-l)7r 



V, V+ — ' V + ' &c. v+—^ — , 



n n n 



it may be shown that their arithmetical mean is equal to 



ao + a„ sin {nv + a„) + a-zn sin (2 nv + a^n) + &c., 

 whatever be the value of v. Hence, as the original series is 

 always convergent, we have, when the numbers is sufficiently 

 great, 



rtO = - {Ul + M2 + Ms + &C. + Un), 



n 

 nearly ; the error having for its limit 



«n + «2n + &c. = ttni nearly. 



Hence, when the period in question is a day, we learn that 

 the daily mean value of the observed element will be given 

 by the mean of two equidistant observations, nearly, when ag 

 and the higher coefficients are negligible; by the mean of 

 three, when as and the higher coefficients are negligible ; and 

 so on. 



The coefficient a^ is small in the case of the temperature ; 

 the curve which represents the course of the diurnal changes 



