MALFATTI. 437 



favourable cases ; by favourable cases we mean cases of non-occur- 

 rence o{ n — 2 white balls. 



By aid of the Lemma in Art. 808 the following equations are 

 immediately established, 



</) (ic + 1, ??) = (f) (x, n - 1), 

 <f> (x-\-l, n — l) — '}f<j> {x, n) + 2 (?2 — 1) ^ (x, n — 1). 



By aid of the first the second becomes 



<f)[x+l,n-l)= n^cf) {x - 1, n - 1) + 2 {n -1) (f) (x, n - 1). 



Thus denoting (^ (x, n — 1) by u^ we have 



«x-+i = i^u^_^ + 2 (?i - 1) u^. 



This shews that ii^ is of the form Aa^ + B/S'^ where a and /3 are 

 the roots of the quadratic 



From the first of the above equations we see that </) (x + 1, n) 

 is of the same /or??i as ^ {x, n — 1); thus finally we have 



</) (ic, w) + </) (x, n - 1) = arj." + h/S", 



Avhere a and h are constants. The required probability is found by 

 dividing by the whole number of cases, that is by ?i^* Thus we 

 obtain 



n 



We must determine the constants a and h by special examina- 

 tion of the first and second operations. After the first oj^eration 

 we must have m - 1 white balls and one black ball in A ; all the 

 cases are favourable ; this will give 



aa + h^ = n^. 

 Similarly we get 



for tlie second operation must either give n white balls in A, or 

 n-1, or 71-2; and the first and second cases are favourable. 



Thus a and b become known, and the problem is completely 

 solved. 



