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



number n is sufficiently great, we may neglect all the terms after the first. 



Hence, approximately, Ao — -'2 (U). 



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

 A„ siu {nx + a„), the succeeding terms being, for the same reason, disregarded 

 in comparison ; and accordingly the limit of error will be A^. 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 2 and the higher coeflScients are negligible ; by the mean of three, when 

 Ai and the higher coefficients are negligible ; and so on. 



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

 variation of temperature ; and, consequently, the ciu-ve which represents the 

 course of this variation is, nearly, the curve of sines. In this case, then, the 

 mean of the temperatures at any two equidistant or homonymous hoxirs 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 temperatirres 

 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^ is small in all the periodical functions with whach 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 vari- 

 ation of the temperature, the atmospheric pressure, and the magnetic declination, 

 as deduced from the observations made at the Magnetical Observatory of Dub- 

 lin 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 corresponding 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 {x -f 239°-0) -1- 0°-70 sin {2x + 67°-2) 

 + 0°-26 sin (3a; -1- 73°-5) + 0°-03 sin (4j; + 102^7) 

 -f 0°-14 sin \bx + 258°-6) 4- 0°.09 sin (6x + 180°). 



