Genesis of the Law of Error. lf>5 



Both these statements cannot be true. But they may both 

 have been thrown out collaterally. For, firstly, the property 

 belongs, with perfect generality, to any number of functions, 

 all the members of the family 



y—\ e- xt cos eddy. * ; 



and, secondly, it does not seem to belong in any obvious, or 

 useful, sense of the term to exp( — h 2 x 2 ) cos (Aw + 7). It is 

 true that the superposition of the last written functions 

 results in convergence to the normal law. Of course it does, 

 if they comply with the fourth condition. And they comply 

 with that condition, since exp( — Ji 2 x 2 ) does; and every 

 element of the integral 



X 



#'exp( — h 2 x 2 ) cos (kx 4- y)dx 

 is less than the corresponding element of the integral 



#*exp( — h 2 x 2 )dx. 



r>oo 



Accordingly no great addition is made to our knowledge 

 when the author concludes : " If we go on piling error upon 

 error, provided each has the fluctuating character indicated 

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

 Gauss " (Phil. Mag. p. 350). But why drag in the " fluc- 

 tuating character"? The proposition remains true when 

 the proviso is omitted. It is true, for instance (for the 

 reason just given), of the function (exp — 7i 2 # 2 )/(l + a? 2 ). 



No doubt there is some interest in verifying the fact of 

 convergence by actual integration. But this interest is not 

 very great when the deviations, extending to infinity, are 

 such as never could occur in concrete nature. The verifica- 

 tions given by Dr. Burton in the Philosophical Magazine for 

 December 1889 are much more interesting. 



The author describes his contribution "as a view by which 

 we can see the law coming into existence, which I submit the 

 other forms of proof, one and all, fail to supply." Is not 

 the desiderated view supplied by regarding the frequency- 

 function pertaining to an aggregate as of the form 

 e x (1 + R), where U is the continuation of Poisson's ex- 

 pansion in ascending powers of 1/ \/n\ and observing, say. 

 on the lines of Morgan Crofton's method, that when a new 

 observation is taken in, n is changed into (n -\- 1) ?t ts the 



* See Oamb. Phil. Trans, vol. xiv. pp.142, IAS (1885). 

 f Todhunter," History of Probabilities," Art, 100:? : Oamb. Phil. Trans. 

 vol. xx. p. 47 (1905). 



