LETTERS TO THE EDITOR 177 



which is attributed to Thiele), namely, that in the notation of the 

 communication the constant c^ is approximately equal to -~ • 



Mr. Fisher definitely states that no criticism of my work is intended, 

 but incidental to bringing out the above relationship he makes certain 

 statements upon which I should like to comment briefly. 



He states that the omission on his part to properly emphasize a 

 close relation between the theory of sampling and the Gram series is 

 probably responsible for the fact that I have intimated that two terms 

 of the Gram series in certain instances yield a better approximation 

 than three or more terms. To my knowledge this is not the case. 



The special form of the Gram series used in my published articles in 

 this Journal is that represented by his Equation 2} The validity of 

 this expansion rests upon the Lebedeff theorem.^ So far as I am aware 

 I have not intimated that two terms of the series yield a better approxi- 

 mation than three or more terms in the sense that 



I F{z) - [c^<pq{z) + ^3(^3(2)] I 

 should be less than 



1 F{z) - [ro(po(z) + r3<^3(2) + • • • + Cn<pn{z)]\ 



irrespective of n, although it is in this sense that Mr. Fisher discusses 

 his example of the graduation of (.9 + .1)""^. To have done so would 

 have been an obvious blunder because, assuming the Lebedeff theorem 

 to be true, the absolute value of the difference e between the function 

 F{z) and the sum of the first n terms of the series can be made as small 

 as we please by taking n sufficiently large. ^ 



I did say, however, in my article in the October issue of this Journal: 

 "Carrying out steps 1 and 2, we conclude that the best theoretical 

 equation representing the data in Fig. 1 is either the Gram-Charlier 

 series (2 terms) or the Pearson curve of Type IV for both of which the 

 estimates of the parameters may be expressed in terms of the first four 

 moments mi, M2, Ms and in of Fig. 3." Of course the first two terms of 

 the Gram-Charlier series requires only yui, /^o and ms- "Best" as used 

 here obviously is in the sense of probability of fit which is entirely 

 different from saying that the first two terms is the best approximation 

 in the sense discussed by Mr. Fisher at least as illustrated by his 



^ It is of course understood that, in practice, transformations are made so tiiat 

 Ci and Ci are both equal to zero. In what follows, therefore, the second term of the 

 series will be £3(^3(3). 



- Fisher, Arne, "Mathematical Theory of Probabilities," 2d edition, 1922, p. 203. 



^ It can be seen from my published work, however, that the sum of two terms is 

 sometimes better than the sum of three. 



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