the Theory of Probabilities. 463 



the probability that a system about to be produced would turn 

 out to be binary, and this would be the value of p. Just as a 

 perfect physiologist and mathematician would be able to assign 

 the probability that an expected birth would be of a given sex. 

 The same value would be obtained for p by taking the actual 

 ratio of the number of binary systems to the w^hole number of 

 systems, supposing an infinite number of instances could be 

 known. Now if we are to put aside all prepossessions and all 

 information dra^-n from other som-ces as to the mode of produc- 

 tion of stars, the value of p is, before observation of the phse- 

 nomena, wholly unknown, and may be an\i:hing fi-om to 1 ; 

 so that within these limits dp is the probability (before observa- 

 tion) that the true value oi p is betw^een j9 BnA. p + dp. 



12. Let Pf* be the a priori probability that in a number of 

 systems comprising altogether n stars, there would be i binaiy 

 systems. This \\Till be a function of n, i and/*, and is calcul^le 

 without difficulty, as we shall see afterwards, supposing p to be 

 known. 



lict Q^ be the a priori probability that out of s single stars, 

 whose configurations wei-e accidental, there would be r paii-s 

 optically double. The calculation of Q^ is a problem invohing 

 only analytical difficulties. 



Then supposing a determinate value for p, the a priori proba- 

 bility of the observed phsenomenon, on the supposition that /u- 

 out of the m double stars are binary, is P^Q"^~^f ■ Let 



the summation being made vnth respect to /x, from /x = to fx.=m. 

 Then putting /^'<S)<^=scj-, the a posteriori probability that the 



value of^ is between j9 anAp + ^ is — -. 



Hence the a posteriori probability that p lies between these 

 limits, and that /jl out of the m double stars are binary, is 



^ J P " G" ~ -'' P " O'*""^'* 

 ^dp '^ijH„_f, _ ^fi^,u-^ , 



CT q? ■or 



and therefore the whole a posteriori probability that /* are binaj-y is 



^X'p;Qr-';* (^.) 



* I have thouj^ht it unnecessary to indicate by any peculiarity of nota- 

 tion, tliat the ii])|)er indices in this and the following expressions are not 

 exponents of powers. 



