350 The Genesis of the Law of Error. 



gress as an approximation more visible ? It is only here that a 

 few remarks in my paper may claim some novelty. Yet so 

 much has been written on the theory of errors that even for 

 these I should not be surprised to find an anticipator. 



If we take a distribution of errors subject to the regular 

 law, that is to say occurring proportionately to exp( — h 2 x 2 ), 

 and replace each element of this by another distribution, 

 also subject to the law, and collect the results, in the order of 

 their magnitude, a third final distribution emerges which 

 again is subject to the law. This is the reproductive property 

 of the law of error, which has been proved apparently by a 

 number of people. By itself such a property leads us 

 nowhere, for its application is limited to domains already 

 subject to the law. Therefore we must travel outside it to 

 find the genesis of the law. But I make two further points. 

 First, if we take a distribution not strictly under the law 

 exp (— h 2 x 2 ), but under one fluctuating about it, say 

 exp ( — h 2 x 2 ).(l + a cos kx) and go through the same ope- 

 ration of disturbing it by a second distribution of the same 

 kind, say under the law exp ( — h' 2 x 2 ).(l-\-a / cos^x), we 

 get a third resultant distribution in which the fluctuating 

 element tends to efface itself. Hence if we go on piling error 

 upon error, provided each has the fluctuating character indi- 

 cated above, we shall as a limit converge to the pure law of 

 Gauss. My other point is that to obtain an approximation 

 to a set of numbers fluctuating about the law of distribution 

 exp { — h?x 2 )i where h is an adjustable constant, nothing more 

 is requisite than to take as originating the error, say for 

 precision, any holomorphic function and then to get the 

 frequency curve register the number of times individual 

 values occur, disregarding at the same time the order in 

 which these values arise naturally. With a single-valued 

 function possessing one maximum and one minimum, the 

 resulting frequency graph will be two portions of the axis of 

 x extending from plus and minus infinity respectively and a 

 portion parallel to the axis between them. For example, a 

 sine curve for the generating function gives such a distribution 

 which may be considered the first rude approximation to an 

 error curve modified by fluctuations. If Prof. Edgeworth's 

 criticism of my treatment of this point implies that when 

 observations are vitiated by the occurrence of a neglected 

 term of the form a sin kt, we should find among them infinitely 

 more cases of occurrence of maxima than of zeroes, as it 

 appears to do, then I beg to differ. I think all values within 



