and Oil the Distribution of the Fixed Stars, 529 



We have then 



p_ _ c(l-a) 



c{l—a) + a{l-py 



First suppose that we had jl>=1, then 



c(l-g) _, 

 c(l-a) 



If, then, it were certain that such a combination as /8 Capri- 

 corni could not exist upon the principle of random distribution, 

 it would be certain from its existence that the prmciple of random 

 distribution did not hold. This is agreeable to common sense. 



Suppose, in the second place, that there was a high a priori 

 probability in favour of the principle of random distribution. 

 Then a approaches to unity, and the value of P diminishes. 

 Hence the probability afforded by the existence of /3 Capricorni 

 against the principle of random distribution diminishes also. 

 And it is perfectly evident from the value of P, that the a priori 

 probability in favour of the principle of random distribution might 

 be so great as to outweigh altogether the influence of the obser- 

 vations made upon /3 Capricorni. This appears also to be quite 

 agreeable to reason. 



3rd. Suppose that there is but a very small probability that 

 /3 Capricorni could result under any other condition than that of 

 random distribution ; the value of c is then very small, and the 

 probability against the hypothesis of random distribution dimi- 

 nishes. If c were 0, i. e. if it were certain that /3 Capricorni 

 could not appear under any other circumstance than that of a 

 random distribution of the stars, then the expression would vanish 

 however near p might appi'oach to unity. However unlikely, 

 then, the principle of random distribution may be, it becomes 

 certain whenever it is known that no other cause can produce 

 the effect observed. 



Let us in\estigate the conditions under which P is greater 

 than pj. If P is greater than p, we have 



extend to different acts of the Creative Power; that it ought to enable us 

 to say with reference to each of them, « hether the connexion of effect with 

 cause was manifest in the ])ha;nomena of stellar distribution ; and that it 

 should make know n to us in Iww many of the particular instances in which 

 such connexion was perceived, the occurrence of double stars was an attend- 

 ant result. There are, however, cases, and perhaps the above is one, in 

 whidi we may with some confidence |)roceed upon other grounds than 

 direct exjierience, e. <j. upon the reasovableness of an h\pothesis viewed in 

 the light of that general analogj' of nature which experience tends daily to 

 confirm. Data derived from tliis soiuce may properly be said to be a priori 

 in relation to jn'oblems such as the above, and it is thus that I have referred 

 to them in the concluding portion of the paper. 



