364 Mr. R. Campbell on the Uniformity 



plicated one, yet it will be easy to obtain a comparatively simple 

 expression for the ratio of the probability of any given number 

 occurring to that of the average number occurring. And it can 

 easily be shown that the ratios of the probability of the number 

 b being the one for a given year to that of b — \, b — 2, b — 3j 

 and so on being the numbers, arc as follows : viz. 



b to {b-1), 



n — b+l ab — b+1 



b an—ab — n — b 



b to {b-^), 



(«-6 + l)(n-ft + 2) {nb-h + \){ab-b + 2) 



b[b — \) (^Q^^(i})^n — b){an — ab—n—b-\-\ 



b to (A -3), 



(n-6 + l)(ft-6 + 2)( n-& + 3) 

 b{b-l){b-2) 



{ab-b + \){ab-b-\-2){ab-b + 3) 



(a,) 



d^il 



[an~ab—n — b){an — ab — n—b + \){an—ab—n—b + 2) 



\ {^3) 



and so on. 



In a similar manner we shall find the ratio of the probability 

 of the number b occurring to that of the numbers b, b + 1, i + 3, 

 &c. . . . occurrino; to be — 



b to {b + l), 



b + 1 an—ab — {n — b — \) 



(«, 



n — b ab — b ' 



bio (^- + 2), 



(& + l)(^ + 2) {an-ab—{ii-b-\)}{an-ab-(ri-b-2)] 

 {n-b){n-b-l) ' {ab-b){ab-b-l) " ^"^' 



bto {b + S), 



{b + l){b + 2){b + S) 



(n-b)[n-b-\){n-b-2) 



{an—ab—{n—b — i)) {an — ab—{n—b—2)][an—ab—{n — b — 2>)] 

 {ab -b){ab-b-\) {(lb -b-2) 



If we suppose n very large with respect to the other numbers, 

 •which is the case of most common occurrence, they will become 

 still more simple. They will then be 



