622 PROFESSOR BOOLE ON THE COMBINATION 



whence 



aJl — c„) xyzws 



^—^a x c lPl = \- . . ( 5 ) 



In like manner we find 



a, (1 — c,) xyzvt 



t^f^^="V (6) 



Again, since we have from the above 



a 2 -a 2 c 2 _ y , a.-a 1 c 1 _ x 



and 



l — a 2 c 2 y + 1 1— a 1 c 1 a?+l 



we have 



(l-« lCl )(l-a 2 c 2 ) (a? + l)(3/ + l) 

 moreover, 



(# + 1) (v + l) z 



r -«l C lPl-«2 C 2-P 2 = - -^ L 



Multiplying the two last equations together we find 



a, (1 — c.) « 9 (1 — c 2 ) . xyz 



NoWj u = xyzws + xyzvt + xyz 



(7) 



Substituting in this expression the values found for its several terms in (5), (6), 

 and (7). we have 



a 2 (l — c 2 ) aJl — c,) a. (1-c,) a'(l — c 9 ) , , 



1 — a 2 c 2 1X l 1 — a x c x l z ^ 2 (1 — a x 0^(1— a 2 c 2 ) v 1 l/a - 2 ^ 2/ 



This is the value of Prob. xyz. That of Prob. an/z will be found by simply changing 

 in the above expressions,^, and r, into l- Pv l-p 2 , and 1-r respectively. These 

 expressions admit of some reductions, and give 



(8) 

 (9) 



*>* 1^ ?1^ Z^i {^ P-») +£fog-ft>+H 



whence we find for the a posteriori value of Prob. z, 



Prob. xyz _ !=% € i *>■ + TZ~ c *P* + r 

 Prob. xij 1 — a, , 1 — a, , 

 1-Cj * l-c 2 2 



Equating this to r we have 



1 — a 1 — a 2 /l — a 1 — a \ 



Whence 



1— a, l— a 



«•= 1-U I-. ; • ... (10) 



r c i+T-^ c 2 



1 1— ^2 



