440 J. Kleiber on " Random Scattering " 



lities,' agreeing apparently with the views of its author. 

 He writes : — 



" The late Prof. Forbes wrote a very interesting criticism 

 on Mitchell's memoir. He objects with great justice to 

 Mitchell's mathematical calculations, and he altogether dis- 

 trusts the validity of the inference drawn from these calcula- 

 tions " (History &c. p. 334). 



Relying on so high an authority, the same paper is referred 

 to by J. Jevons in his ' Principles of Science/ He says : — 



" The calculations of Mitchell have been called in question 

 by the late James D. Forbes, and Mr. Todhunter vaguely 

 countenances his objections, otherwise I should not have 

 thought them of much weight " (Principles &c. vol. i. 

 p. 286). 



Now the article in question contains a very erroneous 

 conception of the " law of great numbers," and its applica- 

 tion to the investigation of accidental distribution, and the 

 objections of Prof. Forbes against this application, together 

 with the experiments made by him on the random distribu- 

 tion of grains thrown on a chess-board, furnish, on the con- 

 trary, a very good argument for and illustration of the very 

 same views they are intended to invalidate. 



Misinterpretations of the laws of " random scattering " 

 like those of Prof. Forbes are not very rare, especially in the 

 writings of statisticians, so I thought it worth while to show 

 the error of these misconceptions. 



It is a common error to confound random scattering with 

 uniform distribution. It is true that the most probable result 

 of a series of drawings from an urn containing an equal 

 number of black and white balls will be an equal number of 

 both ; but this result is in itself very improbable, because this 

 is but one out of a great number of possible events, all giving 

 unequal distribution, less probable individually, but more 

 probable in sum, than the most probable result. Also, the 

 most probable distribution of points on a surface, if scattered 

 at random, is a uniform one, but this is very improbable. 

 The probable result of a random scattering is therefore not a 

 uniform distribution. 



Let n points be distributed on a surface divided into m 



equal parts (squares). Then, although the probable number 



n 

 of points in each square is —, it is very probable that there 



will be squares containing less than — points, while others 

 will contain more. 



Let us find the probable number of squares containing a 

 given number of points. The probability of a given point 



