DISCUSSION OF BAROMETRIC OBSERVATIONS. 471 



these points the curve maintains a nearly uniform height, having its 

 minimum on January 1, when the vahie is 0.2254:. The minimum for 

 the year is 0.1089 on July 23. 



Besides its significance as an independent meteorological element, the 

 function may also be regarded in its relation to the probable error of 

 the numbers composing Tables II, III, and IV. Eeferring to the method 

 by which the mean departures were computed it will be seen that as the 

 number of daily means in excess of normal value varies but little from 

 the number in defect, the inaccuracy of the normal value of i)ressure as 

 derived from Table I must be too slight to affect appreciably the result- 

 ing formula for mean departure; hence this mean departure may be 

 regarded as identical with the mean of the errors of single daily means, 

 regarded as measurements of the true normal pressure. The " mean 

 error" will then be found by multiplying the mean departure by the 

 constant 1.2533, or the probable error by multiplying it by 0.8453. Ap- 

 plying the latter factor to the extreme and mean values of the function 

 it appears that the probable error of a single daily mean is ±0.1923 on 

 February 4, ± 0.09205 on July 23, and has a mean value of ± 0.1500. If 

 no daily means had been omitted in the records each of the numbers in 

 Table II would have been derived from twenty-five daily means, and its 

 ]irobable error would have been one-fifth that of the daily mean. But 

 in tact 184 of the 9,131 daily means are wanting, so that the average 

 number of means in a column is 24.496. Hence the factor by which the 



X>robable error of a dailj^ mean is to be multiplied is— 7—=^ or 0.202. 



v24.496 



The products, .0388, .0183, and .0303, are the extreme and mean values 

 of the probable error of the numbers in Table II. Each of the numbers 

 in Table III is the mean of 10.15 numbers of Table II, hence the prob- 

 able errors of the values in Table III are: Maximum, .0122; minimum, 

 .00o8 ; mean, .00951. 



Finally, Table IV is derived from Table II by taking the tenth order 

 of means. Now, if from the series of terms Ti, T2, T3, etc., m successive 

 orders of means be derived {m being an even number), the value of 

 that term in the mth. order which corresponds to Tn is : 



IW +mT ,)n{m-l)rj^ , 



+'''^ -^ (»-)-iMi-l) +-•- (n + *?n) p 



the coetficients being those of the mth. power of a binomial. 



Hence, supposing the probable error of each term of the original 

 series to be r, the probable error E of a term of the mth order will be : 



E=|,.|l+»'-+^(|=l)V....+,„Hlp- 

 If 10 be put for m in this formula, we have 



E=0.4198r. 



