LETTERS TO THE EDITOR 173 



The omission on my part to properly emphasize the close relation 

 between the theory of sampling (i.e., the a posteriori probability theory) 

 and the Gram series is probably also responsible for the fact that Dr. 

 Shewhart in several of his articles has intimated that two terms in the 

 Gram series in certain instances yield a better approximation than three 

 or more terms. This idea has probably arisen from the mistaken 

 notion on the part of Bowley of the generalized probability curve, 

 which is a special example of the general Gram series. The following 

 brief remarks should, therefore, not be taken as a criticism of Dr. 

 Shewhart's work, but rather as a sort of amplification of some of the 

 chapters in my own book on "The Mathematical Theory of Proba- 

 bilities." 



Gram's series, like the Fourier series, offers a perfectly general 

 method for the expansion of arbitrary functions and is, contrary to the 

 opinion of some students, not limited to frequency functions, although 

 it there happens to be especially useful. 



The underlying principles of the Gram series may be set forth 

 briefly as follows: Let F{x) be the true (or presumptive) function, 

 which is known from either purely a priori considerations, or from 

 observations, and let G{x) be another function (the so-called generating 

 function), which gives a rough approach to F{x). Then according to 

 Gram's method, we have 



F{x) - c,G{x) + c,G'{x) -f c,G"{x) + . . . + r„G«(x). (1) 



The generating function G{x) may assume a variety of forms. In 

 the case of generalized frequency functions, it is customary to select as 

 the generating function, G{x), a quantity z = h(x) which is normally 

 distributed, and write F{x) as ^ 



F{x) = CoiPi){z) + riv?i(s) + C2(p-2(z) + . . . -f Cn<Pn(z), (2) 



where ^u(s) = , — e~^''- is the generator and <pi{z), ^2(2) • • • <Pn(z) 



■\2ir 



its derivatives. 



When viewed from the theory of elementary errors as originally 

 introduced by Laplace in his monumental work, "Theorie des Proba- 

 bilities," the Gram series takes on special significance in the way in 



^ Ifz = h{x) = (x — M): a, or a linear function of .v, and if the origin of the 

 co-ordinate system is laid at M with a as its unit, we have the special case, or the 

 Charlier A series of the well-known form 



F(x) = N[<po{2) + Pzfziz) + &i<p,{z) + ...]. 



The various types of the frequency curves of Pearson may of course also be used as 

 generators in the Gram series. 



