178 BELL SYSTEM TECHNICAL JOURNAL 



example. In this case I found that the probabiHty of fit for two terms 

 was greater than that for three. Now, I find that it is as good as for 

 Mr. Fisher's third approximation. It may be of interest also to know 

 that statistical distributions sometimes arise where the first three 

 terms give as good a fit as Mr. Fisher's third approximation involving 

 4 terms. This is particularly true when the universe from which the 

 sample is drawn is nearly symmetrical. My action in this connection 

 can be justified both upon theoretical and practical grounds but we 

 need not do more than mention this point to make sure that the 

 reader will not confuse my statement quoted above with what Mr. 

 Fisher is talking about in his communication. 



Having thus dismissed the questions which may arise in connection 

 with published work in this Journal, I should like to add a word or two 

 of caution to the reader of Mr. Fisher's letter where it reads: "More- 

 over, since extensive tables, notably those of Jorgensen, now are 

 available for the normal function and its first six derivatives, there 

 seems no good reason why we should not use the more exact approxi- 

 mation than the inexact formula by Bowley." 



We have made far more use of the Gram series in connection with 

 our inspection work than indicated in the published papers. In 

 this work we have found that it is theoretically not necessary in 

 certain instances and in many more instances it is not practical to 

 follow Mr. Fisher's suggestion. I shall limit my remarks to the 

 application of the series which we have made in expanding a known 

 function in terms of an infinite series in which the generating function 

 is the normal law. In this connection the outstanding practical 

 question is: Given the known function F{x), what number n of 

 terms of the infinite series must we take in order that the absolute 

 magnitude of the difference between the function Fix) and the sum of 

 the n terms will be less than a given preassigned quantity e? I am 

 sorry that Mr. Fisher does not answer this question. Instead he 

 proposes a grouping of terms upon the basis suggested in a footnote 

 to his article. Now, it may easily be shown in the particular case 

 cited by Mr. Fisher, i.e., the graduation of the point binomial (.9 + .l)^'"', 

 that the sequence of signs depends upon the value of z, that for certain 

 values of z his second approximation is just as good as his third, and 

 that in many instances the difference between the second approxi- 

 mation and the third is not sufficiently great to be of any practical 

 importance. Whether we should use the second, third, or higher 

 approximation in a given case is one for special consideration. 



In closing let me say that I have not made the above remarks with 

 any intention of discrediting the applications of this series but rather 



