236 ON THE MEAN RESULTS OF OBSERVATIONS. 



of the series, -4 w sin (nz + a n ), the succeeding terms being, for the 

 same reason, disregarded in comparison ; and accordingly the 

 limit of error will be A n . Thus, 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 observed values, 

 nearly, when A* and the higher coefficients are negligible ; by the 

 mean of three, when A 3 and the higher coefficients are negligible ; 

 and so on. 



4. The coefficient A 2 is small in the series which expresses the 

 diurnal variation of temperature ; and, consequently, the curve 

 which represents the course of this variation is, nearly, the curve 

 of sines. In this case, then, the mean of the temperatures at 

 any tico equidistant or homonymous hours is, nearly, the mean 

 temperature of the day. The same thing holds with respect to 

 the annual variation of temperature ; and the mean of the 

 temperatures of any two equidistant months is, nearly, the mean 

 temperature of the year. These facts have been long known to 

 meteorologists. 



5. The coefficient A 3 is small in all the periodical functions 

 with which we are concerned in magnetism and meteorology; 

 and, therefore, the daily and yearly mean values of these functions 

 will be given, approximately, by the mean of any three equidistant 

 observed values. 



In order to establish this, as regards the daily means, I have 

 calculated the coefficients of the equations which express the laws 

 of the mean diurnal variation of the temperature, the atmospheric 

 pressure, and the magnetic declination, as deduced from the 

 observations made at the Magnetical Observatory of Dublin 

 during the year 1843. The observations were taken every 

 alternate hour during both day and night; and the numbers 

 employed in the calculation are the yearly mean results corre- 

 sponding to the several hours. The origin of the abscissse is taken 

 at midnight. 



6. The following is the equation of the diurnal variation of 

 temperature : 



U- A, = + 3-60 sin ( * + 239-0) + 0-70 sin (2x + 67-2) 

 + -26 sin (3a? + 73 -5) + -03 sin (4x + 102 -7) 

 + -14 sin (5* + 258 -6) + -09 sin (6* + 180 -0). 

 Hence, the error committed in taking the mean of the temperatures 



