618 PROFESSOR BOOLE ON THE COMBINATION 
The term «ywvs t, reduces to xywv, and represents a class some part of 
which is included, and some part not included under the class «yz, 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 2 would, on effecting 
the above-named substitutions, become 0, indicating the absolute non-existence of 
the class which it represents. 
Resuming the value of ¢, and adopting the simplification of Art. 21, we find 
for V the value 
V=ayws+ vyvitacws+yut+etyt l+aywt+aw+ ayutyv+ay 
=(@+1) (y+) +yv(e@t)) +1) +awyt+)(st+1) . ; (15) 
And hence we have the following system of algebraic equations: 
a(y +1)4+ vyu(t+1)+ew(y +1) (s+)) y(a +1) +yu(@ +1) (¢+1)+awy(s +1) 
a, Ay 
_ru(y +1) (s+1) _ yro(e+)) (+1) _ wew(yt+l)s _ yo(w+1)t 
Lake Oy Cy Aye Py Dy Co Po 
= Sys FAY Y (w + I(yt1) + yet) (t+) +awytl)(e+1) (16) 
From these equations, if we assume 
(+1) (y+])+yu(@+)) (+1) + aw(y+1)(s+D=A, 
d being a subsidiary quantity introduced for convenience, we readily deduce 
ee +ayvtt+ cay 
(17) 
aws(y+1 
a op = yD 
ae Ter a ean oe 1) 
1 _(@+1) (41) +eu(y+1) (s+) 
Raa UK 
a,¢,p,xa,(1—e,) _  ayws 
Hence cars Sate een hate an Nem) 
Tn like manner 
Ay Cy Pox a,(l1—¢,) _ awyvt 
1-a,¢, = by ; : i : (19) 
Again we have 
a(y+1)+ayu(t+1 
a,(1—¢,)= (y+1) wot ( ) 
PCC Er ie ss 
