between Stars forming Binary or Multiple Groups. 427 



to be both 99. But this event can happen but one way; that 

 is, by each die showing the number 99. Consequently the 

 probability for this second event is 



1 1 1 



100 x 100 x P ; 

 or but half that for the former. 



The same result is obtained from the general formula of 

 Note A, applied to find the chances of no duplication and of 

 one duplication when ft = p=100. 



Postscript. 



In finally sending this paper to the press, after several mo- 

 difications and additions, I may be permitted to express a 

 hope that its arguments may be weighed with deliberation, 

 and that first impressions, however strong, of a contrary ten- 

 dency may be distrusted. The extreme generality of the con- 

 siderations which enter into the discussion renders it difficult 

 to lay down propositions sufficiently large to include them, 

 and yet free from the risk of error through that very gene- 

 rality. I would therefore invite attention to the particular 

 conclusions which I negative, and which must be maintained, 

 I conceive, by any one who is disposed to adopt the views of 

 Mitchell and his followers : — (1.) That there is any calculable 

 probability, such as 9570 to 1, against the observed occurrence 

 of 2 stars out of more than 10,000, within 4" of one another, 

 having been fortuitous (Arts. 5, 40). (2.) That the fact of two 

 stars being seen within an infinitely small distance of each 

 other, amounts to a mathematical proof of the certainty of 

 their being physically connected (Art. 30). (3.) That were 

 the stars uniformly spaced over the heavens or arranged with 

 perfect symmetry, no argument could be alleged against such 

 arrangement being the result of chance, but any deviation 

 from symmetry would raise such an argument (Arts. 30, 31, 32). 



I have now devoted a great deal of time and attention to 

 the consideration of this subject. I have profited too by the 

 kind criticisms of several friends, to whom I submitted the 

 argument. 1 do not think that my opinions are likely to 

 change; and other engagements now require all my attention. 

 Should this paper call forth any controversial remarks, I hope 

 I may be excused (for the present at least) from entering fur- 

 ther into the discussion without the imputation of want of 

 courtesy. Silence, instead of implying consent, ought in such 

 a case to signify the reverse; for, should I be satisfied that 

 I am wrong in any point, I should in that case feel it a 

 duty to acknowledge it. 



Edinburgh, 1 6th November, 1850. 



