the Theory of Probabilities. 465 



probability that the uumber of couples drawn was /i ? /li being 

 any assigned number not greater than -n. 



With reference to the astronomical problem, it must not be 

 forgotten that a binary system cannot have been formed acciden- 

 tally out of two single systems, at least if we limit the term to 

 the case of two stars describing elliptic orbits of moderate dimen- 

 sions about then- common centre of gravity ; since it is dynami- 

 cally impossible that two stars, acted on only by their mutual 

 attraction, could be brought into such a relation with one another 

 if they had ever been separated by an intei-val greater than the 

 aphelion distance of the relative orbit. We are therefore not de- 

 parting much from the facts of the case in assuming that a 

 binary system must have been originally produced as such. We 

 neglect only the possibility of two single systems having been 

 united into a binary system by some distinct interfering cause. 

 It is true, however," that we have neglected a circumstance, less 

 important in principle, though jirobably more so as aflfecting 

 numerical results, namely, the possible or actual existence of 

 multiple systems. But to take account of this would have intro- 

 duced extreme complexity, without adding anything to the com- 

 pleteness of the investigation so far as illustration of principles 

 is concerned. 



15. Let us now return to the expression (yu,) art. 1.2, and con- 

 sider the function Qf.. 



The a priori probability that two given stars, whose positions 

 were accidental, would be within a given angular distance 9 of 



one another, is sin® ^. About this there is no difficulty or dis- 



pute ; and the expression is independent of any hypothesis as 

 to their respective distances from the ej'^e. From this it is theo- 

 retically possible to calculate the a priori probability that s single 

 stars would be accidentally so grouped that there would be r 

 such pairs, and that with these exceptions no t\;'o stars would be 

 within a given angular distance 0. But the problem is, in its 

 rigorous form, very difficult, if not entirely impracticable. It 

 becomes comparatively simple, however, if we admit, as a suffi- 

 cient approximation, the representation of it given (or suggested) 

 by Professor Forbes. 



Suppose s dice, each having t faces, where t is the whole 



number nearest to 3, the faces being numbered from 1 to t. 



e ^^ 



Then sin® 7' is the probability of throwuig doublets with a given 

 pair of dice ; and the approximation in question consists in 



