SECT. 4.] Randomness and its scientific treatment. 99 



the horizontal being equally represented in the long run. 

 We have therefore to contemplate a surface of uniform dis 

 tribution, but it will be the surface, not of the ground, but 

 of a hemisphere whose centre is occupied by the man who 

 iires. The ultimate distribution of the bullets on the spots 

 where they strike the ground will not be uniform. The 

 problem is in fact to discover the law of variation of the 

 density of distribution. 



The above is, I presume, the treatment generally adopted 

 in solving such a problem. But there seems no absolute 

 necessity for any such particular choice. It is surely open to 

 &ny one to maintain 1 that his conception of the randomness of 

 the firing is assigned by saying that it is likely that a man 

 .should begin by facing towards any point of the compass in 

 differently, and then proceed to raise his gun to any angle 

 indifferently. The stage of ultimately uniform distribution 

 here has receded a step further back. It is not assigned 

 directly to the surface of an imaginary hemisphere, but to 

 the lines of altitude and azimuth drawn on that surface. 

 Accordingly, the distribution over the hemisphere itself will 

 not now be uniform, there will be a comparative crowding 

 up towards the pole, and the ultimate distribution over the 

 ground will not be the same as before. 



4. Difficulties of this kind, arising out of the un 

 certainty as to what stage should be selected for that of uni 

 form distribution, will occasionally present themselves. For 

 instance : let a book be taken at random out of a bookcase ; 

 what is the chance of hitting upon some assigned volume? 

 I hardly know how this question would commonly be treated. 

 If we were to set our man opposite the middle of the shelf 



1 As, according to Mr H. Godfray, proposed in an examination. See 

 the majority of the candidates did the Educational Times (Eeprint, Vol. 

 assume when the problem was once vn. p. 99.) 



72 



