514 Mrs. S. Bryant on the Deduction of Inductive Principles 

 (r) n—r const, may be x, and r constituents x ; 



No. of ways = 



r n — r 



(n) const, may be x, and n constituents x ; 

 No. of ways = 1. 

 Thus, the number of possible constitutions 



O-i) 



= 1 + ^ + 



1.2 



+ ... + n + l = 2 n ; 



the number of types = the number of terms in this expansion 



= n + l; 

 the numbers of each type = the respective terms 



= coefficient of x n ~ r x r in the expansion of (x + x) n . 

 All this may be expressed as follows: — 



Sum of possible universes =2(1) = (x + x) 



7t (n — 11 

 = x n + nx n - 1 x + — ) 9 ; x n 

 JL • L 



-2^2 



X 2 + ... + X n . 



Or thus, reading columns vertically and adding, with the 

 understanding that 1 denotes presence of x, its absence, and 

 t\)tty &c. successive times in the Time- Universe: — 



h 



2(1)= * 8 



1010101010101010 

 1100110011001100 

 1111000011110000 



111111111111 



00000000 



Now, if all constitutions of the Universe are equally pro- 

 bable, the probability of the occurrence of any one type is the 

 frequency of its occurrence in the above divided by 2 n ; and 

 the various probabilities thus obtained are those which mathe- 

 maticians have assumed to be equal. We may denote each 

 type as of the form x m x n ~ m , the relative frequency of the 



?n 



event x in such a universe being — y and the relative frequency 



w 





of that type being 



2 n n — m \m 



We have, then, 



