354 Mr. A. Cayley on a Question in the 



p. 268, 1855. In fact, writing for shortness a' = l— «, &c, we 

 have 



U — Cip— fiot'/JL, 



and thence 

 which gives 



/3' + /3q-u=/3'{l-u\), 

 u' + aq — w = a'(l— /3/x), 



that is, 



^-^(a' + ^ + ^ + ^-a^ + ^' + a^^ + ^-a^^Oj 

 or, what is the same thing, 



u*--u{l--otj3 + oip + /3q)u + a. t l3q + aj3 ! p + ci/3pq = 0', 

 and thence 



U = l{l-uj3 + ap + /3q-p), 

 where 



p 9 = (1 -«£ + *p + p q )*-4t{*fSp + *7?g + *&??)? 

 which may also be written 



p 2 = (a' - a/3' + ajtf - ffqf + 4aa'/3'/, 

 = (£'- £«' - «;» + j3q)* + 4$3'«y, 

 = (1 -aP + ap-PqF-luPip-eq), 

 = (1 -ajS-ctp + Pq^-^Piq-up); 

 and we then have 



^^=1(1-^-^ + ^-^, 

 j3'a\=i(l-*0 + *p-l3q-$). 



As probabilities, a, /3, jo, # are all of them positive and less 

 than unity (so that a', /3', jt/, #' are all positive) ; u, X, //,, which 

 are also probabilities, must likewise be positive, and less than 

 unity : and in order that this may be so, it is necessary and suf- 

 ficient that 



p<$q, q< *p, 



and that p shall denote the positive square root of the above- 

 mentioned value of p 1 . The solution is therefore inapplicable, 

 unless the data are such that 



# p <£/3q, q <fc up. 



It may be added that, the values of X, p, being known, the 

 solution gives the probabilities of all the compound events ABE 

 &c. : thus 



