1884.] President's Remarks on Rainfall. 167 





mean 



maximum 



The Punjab 



... 5 per cent. 



7 per cent. 



N. W. Provinces 



2 „ „ 



3 i» » 



Behar 



... 2 „ „ 



2 » ,» 



Lower Bengal 



... 2 „ n 



^ n » 



" These data suffice only to give some general idea of the amount of 

 error inherent in the figures dealt with by Mr. Pearson, but I must 

 confess that in my mind they engender some misgiving. 



u There is another point of view, from which the results may perhaps 

 be called in question, viz., the method of smoothing ; by which the cyclical 

 variation is determined, the values of which serve as Mr. Pearson's points 

 of departure, for the determination of those residual values, which form 

 the subject matter of his discussion. The method of smoothing adopted 

 is to add to the differential figures of each year, half the sum of those for 

 the preceding and succeeding years, and to divide the sum by two. 

 This method is in accordance with the formula 



(n - 1) (n - 2) 

 a + (n - 1) b ■ + 2 i-i 1 



V = 



1 + (.-!) + &i-l)(-*) 



1.2 



where a, b, and c are the unsmoothed values for three consecutive years, 

 b' the smoothed value for the middle year of the series, and n (here = 3) 

 the number of years in the series. The selection of a series of 3 years is 

 of course arbitrary ; and I have therefore computed a smoothed series in 

 which n = 11 (the full series of years), the numerical values of the serial 

 coefficients of the numerator being 



1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 

 and the divisor their sum. The residues which result when the 

 smoothed values, thus computed, are deducted from the original values, 

 exhibit relations very similar to those given by Mr. Pearson, and there 

 is therefore no reason to attribute those relations to the fact of his arbi- 

 trary selection of a series of 3 years. 



" This method of smoothing, I may explain, is based on the theory 

 of the probability of errors. The fundamental assumption is that the 

 value for any given year of a series, as resulting from the undisturbed 

 operation of a simple cyclical law, may be displaced, owing to the 

 operation of foreign unknown causes, by 1, 2, or more years, with a re- 

 lative probability, proportional to the ordinates of the curve of probabili- 

 ty ; the time intervals being the corresponding abscissce.* 



* A rejoinder to Mr. Blanford's remarks, by Mr. A. N. Pearson, will appear in 

 the Proceedings for December, 



