of Points on a Surface. 441 



square i 

 not being in this square is 



being in a given square is — ; the inverse probability of its 



i-l. 



m 



Hence the probability that this square contains i points and 

 no more is 



! 



n\ my iy-' 



Pi il{n-i)!\m)\ m) ; 



and the probable number of squares containing i points will 

 be mp v or 



m . T _^_(iY( 1 _iy- i . . . . (1) 



i 1 (n — i) ! \m/ \ m/ 



We will apply these formulas to the experiments described 

 by Prof. Forbes. He writes : — 



" I have thought it worth while to test a little by simple 

 experiment the differences to which l mere chance ' gives rise 

 in the grouping of bodies dispersed over a surface, by a 

 method of ' random scattering,' which I conceive to be as 

 nearly as possible analogous to Mitchell's idea of chance as 

 affecting the placing of the stars. ... I placed a chess-board, 

 having, as usual, sixty-four squares, on the floor, and I pro- 

 vided a large sieve into which I put a quantity of grains of 

 rice, which did not fall through the sieve until it was some- 

 what shaken. I then shook the sieve at a considerable height 

 above the chess-board until it was pretty well scattered over 

 with grains. . . . The following diagrams contain the results 

 of five experiments, the number of grains which fell on each 

 of the sixty-four squares being counted and registered. . . . 

 In these experiments we observe that the most loaded squares 

 contain from nearly two to more than four times the average 

 number of grains, whilst in four out of five experiments one 

 or more squares were vacant. ... If we were to take any 

 one of these experiments, and attempt to calculate the ante- 

 cedent probability of the grains so arranging themselves on 

 Mitchell's supposition, we should unquestionably find nume- 

 rical chances far greater against these configurations being 

 the result of accident, than those on which we are told that 

 we form our most certain ordinary judgments. Thus, if an 

 experimental argument may b-3 admitted, the reasoning of 

 Mitchell and his followers is altogether fallacious." 



But if we calculate the probable number of squares con- 

 Phil. Mag. S. 5. Vol. 24. No. 150. Nov. 1887. 2 G 



