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m—On the Mean Eesidts of Observations. By the Rev. Humphrey Lloyd, D. D., 

 President ; F. R. S. ; Hon. F. R. S. E. ; Corresponding Member of the Royal 

 Society of Sciences at Gottingen; Honorart/ Member of the American Philoso- 

 phical Society, of the Batavian Society of Sciences, and of the Society of Sciences 

 oftlie Canton de Vaud, tfc. 



Read June 12, 1848. 



1. 1 HE 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 solu- 

 tions 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 generahty is not 

 eslabUshed. It is the object of the following investigation to supply this defi- 

 ciency, and to show in what manner the daily and yearly means may be ob- 

 tained 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, approximately, by taking the anVAmeftm/ 

 nmn of any number of equidistant observed values ; the degree of approxima- 

 tion, 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 difierent manner; and Gauss has given a me 

 thod, whereby the values of the integral, ["udx, may be obtained with still 

 greater accuracy from the observed values of the ordinate, Z7, corresponding to 



