Letters to the Editor 



From Mr. Arne Fisher : A Relation Between Two Coefficients in the 

 Gram Expansion of a Function 



From Dr. W. A. Shewhart: A Reply 



From Mr. Fisher: A Further Note 



To Die Editor of the Bell System Technical Journal: 



In a number of valuable and interesting contributions to this 

 Journal, Dr. W. A. Shewhart has made an extended use of the infinite 

 series of Gram. With all the controversy that at present is going on 

 between the pure empiricists, attempting on the one hand to dragoon 

 statistical analysis into a mere inductio per simplicem enumerationem, 

 and the a priori theorists on the other hand, who claim that statistical 

 methods so-called are nothing more than simple and evident appli- 

 cations of well-known principles of the probability calculus as formu- 

 lated by Laplace, it has been a source of satisfaction to me to note that 

 Dr. Shewhart apparently has given the latter methods a place of 

 preference over the methods of the out and out empiricists. 



Because of the fact that I happen to be responsible for having called 

 the attention of English-speaking readers to the series of Gram and to 

 have emphasized that Gram's development anteceded the less general 

 developments by Edgeworth and the very special formula by Bowley 

 by more than 20 years, I hope that I may be afforded an opportunity 

 through the medium of your Journal to point out in brief form a few 

 decidedly simple features of the Gram series which greatly add to its 

 practical applications in statistical work. 



Moreover, it seems that Dr. Shewhart, as well as other students in 

 this country, have received a somewhat different idea about the nature 

 of the Gram series than that which it was my intention to convey in my 

 book on "The Mathematical Theory of Probabilities." This probably 

 is my own fault. For while I have given in the above-mentioned book 

 a description of the various methods for determining the coefhcients of 

 the individual terms of the Gram series, I did not mention the various 

 degrees of approximations according to the number of terms as retained 

 in the series itself. The reason for this omission is due primarily to the 

 fact that I expect to treat this aspect in a forthcoming second volume 

 of the book on probability in connection with the presumptive error 

 laws of the a posteriori determined semi-invariants, which laws contain 

 as a special case the evaluation of the standard (or probable) errors of 

 the constants of the frequency curves. 



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