362 I\Ir. R. Campbell on the Uniformiiy 



any given year, and Ag not doing so, is 



b/- ab-l \ 

 n\ an—lJ' 



Now suppose the persons alive in any given year to be Aj, 

 Aj, . . . A„, it will easily be seen from the above that the proba- 

 bility of the phfenomenon in that year being presented by Aj, 

 and neither by Ag, A3, . , . nor A„, is 



b/ r/^> — l y ab—l\ / ab — 1 >. 



n\ an — lA «n — 2/ ' " \ an—7i\-lJ' 



Hence the probability of one person only presenting the pheeno- 

 menon in the given year is 



In a similar way it may be shown that the probability of Aj, 

 Ag both presenting the phtenomenon in the given year, and 

 neither Ag, A4 ... nor A„ doing so, is 



6 ab-l 

 n an 



-IV an-^JV an-z)"'\ an-n + lJ' 



And the pi'obability of two persons onlij presenting the phseno- 

 menon in that year is 



n—l ab-l f ab-2 \f-, ab-2 \ f ab-2 \ 



2 m^\ an-'zA an-sJ" '\ an-n + lJ' ^^ 



The probability of three persons only doing so will be 



{n — l){n—2) ab | 1 ab-2 /' ab-S \r-. ab~S \ 

 2.3 * ' a7i — l a7i—2\ an — 3J\ an—^J 



...(1- ;'-\ ) (3) 



V ab — n + lJ ^ ' 



It is easy therefore to get the general formula, but it is only 

 necessary to write down the most important one, which is the 

 probability of b jjersons exactly presenting the phsenomenon in 

 the given year. This is 



{n-l){n-2)...{n-b + l) ^ ab-l ab-2 ab-b + l 

 2.3...b ' 'a7i—lan—2"'an—b + l 



(i_^yi__f^y..(i__^^Y . (B) 



\ an — b/\ an — b — ij \ an — n-{\./ 

 This expression would be obtained from that for the proba- 



