618 PROFESSOR BOOLE ON THE COMBINATION 



The term xywv~s~t, reduces to xywv, and represents a class some part of 

 which is included, and some part not included under the class xyz, whence the 

 coefficient ~ ; for an event included under the former class may or may not be 



included under the latter. 



Lastly, any term whose coefficient in the expansion is ^ would, on effecting 



the above-named substitutions, become 0, indicating the absolute non-existence of 

 the class which it represents. 



Resuming the value of (p, and adopting the simplification of Art. 21, we find 

 for V the value 



V = xyws + ccyvt + xws + yvt + x + y + l + xyw + xw + xyv +yv + xy 



=(x + l)(y + l)+yv(x + l)(t + l) + xw(y + l){s + l) . . (15) 



And hence we have the following system of algebraic equations : 



x(y + 1) + xyvjt + 1) + xw(y + 1) (s + 1) y (x + 1) +yv(x + 1) (t + 1) + xwy(s + 1) 



«1 «2 



_ xw(y + l)(s + l) _ yv(x + 1) {t + 1) xw(y + l)s yv(x+l)t 

 a x c x a 2 c 2 «iCiPi <*2 C *P3 



= ^^^^±^-=(x + l)(y + l) + 7jv(x + l)(t+l) + xw(y + l)(s + l) (16) 



From these equations, if we assume 



(x + l)(y + T) + yv(x + 1) (t + 1) + xw(y + 1) (s + 1) = \, 



\ being a subsidiary quantity introduced for convenience, we readily deduce 



xyws + xyv t + cxy ._ ,_. 



u =~ x ^ ' 



xws(y + 1) 



«i c iPi= — -x — 



_ (x + l)(y + l)+xw(y + l)(s + l) 

 i— « 2 c 2 — r 



Hence 



a i c i Pi x a 2 (^ ~~ c i) ocyws 

 1 — a 2 c 2 A 



In like manner 







a 2 c 2 p 2 xa 1 (l — c 1 ) xyvt 

 1 - a x c x \ 



Again we have 





(18) 



(19) 



n . x(y + l)+xyv(t + l) 

 «i(! - «i) = — x " 



,.. N y(x + l) +xwy(s + T) 



