IX. ON THE MEAN RESULTS OF OBSERVATIONS. 



Transactions of the Royal Irish Academy. Vol. XXII. 



1. THE problem in which it is sought to determine the daily mean 

 values of the atmospheric temperature or pressure, from a limited 

 number of observed values, is one of fundamental importance in 

 meteorology ; and, accordingly, many solutions of it have been 

 proposed by meteorologists. These solutions are derived, for the 

 most part, from the known laws of the diurnal variation of these 

 elements. Many of them are accordingly applicable only to the 

 particular cases considered ; while for others, which are really 

 general in their nature, that generality is not established. It is 

 the object of the following investigation to supply this deficiency, 

 and to show in what manner the daily and yearly means may 

 be obtained in all the periodical functions with which we are 

 concerned in magnetism and meteorology. 



2. It is known that the mean value of any magnetical or 

 meteorological element, for any day, may be obtained, approxi- 

 mately, by taking the arithmetical mean of any number of 

 equidistant observed values ; the degree of approximation, of 

 course, increasing with the number. A somewhat more exact 

 mean may be deduced, as has been shown by Cotes and Kramp, 

 by combining the equidistant observed values in a different 

 manner ; and Gauss has given a method, whereby the values of 



the integral, [*" Udx, may be obtained with still greater accuracy 



J- 



from the observed values of the ordinate, U, corresponding to 

 certain definite abscissae.* But in the case of periodical functions, 



* Commentationes Socutatis Reyia Scicntiaruin Goltinyensis, torn. iii. 



