ON THE MEAN RESULTS OF OBSERVATIONS. 237 



at any two equidistant hours as the mean temperature of the da}% 

 is expressed nearly by the term 



0'70sin(2#+67-2); 



and, consequently, cannot exceed 0'70. To obtain the pairs of 

 honionymous hours, whose mean temperature corresponds most 

 nearly with that of the day, we have only to make sin (2* + 67 -2) = 0, 

 which gives for x the values 



x = 56-4, x = 146-4, 

 corresponding to the times 



t = 3 A 46"', t = 9* 46"'. 



Accordingly, the best pairs of homonymous hours, so far as 

 this problem is concerned, are 3* 46'" A. M. and 3 7 ' 46"' p. M., or 

 9*46 M A.M. and 9* 46 m P.M. 



The error committed, in taking the mean of the temperatures 

 at any three equidistant hours as the mean temperature .of the 

 day, is, very nearly, 



andean not, therefore, exceed 0> 26. The best hours are those in 

 which the angle, in the preceding expression, is equal to 180 or 

 360. The corresponding values of x are 



x = 35-5, x = 95-5 ; 

 whence 



t = 2 h 22'", t = 6 A 22'". 



Accordingly, the best hours of observation are 



2 h 22"' A. M., 10 A 22'" A. M., 6* 22'" P. M. ; 

 and 



Q k 22'" A.M., 2 h 22'" P.M., 10'' 22'" P.M. 



By taking the mean of any four equidistant observed values, 

 the limit of error will, of course, be less. Its amount, which is 

 the coefficient of the fourth term of the preceding formula, is 

 only 0-03 ; and, accordingly, the mean temperature of the day is 

 inferred from the temperatures observed at any four equidistant 

 hours with as much precision as can be desired. 



