612 PROFESSOR BOOLE ON THE COMBINATION 
those, which contain ¢ as a factor, &c.; and then regarding s, ¢, &c., as algebraic 
quantities. From the system thus formed, we must determine w as a function of 
p,q . . and the arbitrary constant c, should it be present. This will be the solu- 
tion of the problem. 
The quantities s,¢ . . are the same as p’,q’ . . and represent the probabilities 
of the hypothetical simple events, represented bys’, ¢. . Accordingly, as pro- 
babilities, they must admit of being determined as positive proper fractions, and 
that the solution may not be ambiguous, they must admit of only one such de- 
termination. These conditions will be fulfilled whensoever the problem represents 
a possible experience, and it will be then only fulfilled. And in this way, or by 
directly investigating the conditions of possibility by the rule of Art. 14, a solution 
is made determinate. 
The arbitrary constant ¢ does not, as has been intimated, always present itself. 
When it does, it represents the unknown probability, that if the event C occur, » 
will occur. It indicates, therefore, the new experience which would be necessary 
in order to make the solution definite. 
18. I will, for the sake of illustration, apply the method to the problem of 
Art. 11, and in so doing I will limit the solution by the conditions relative to 
s, t, &e. 
The problem, as symbolically expressed in Art. 13, is as follows :— 
Given Prob. zy=p Prob. yz=q Prob. zv=r } (1) 
Required Prob. xy 
Translating the problem as directed in the first part of the rule, we write 
8 Z=t 2u=U 
ig ee: } Pecith . 2) 
whence, by the calculus of logic, 
w=stut+0(stut+ttsutustt+svt) 
i ete oS = 
+9 (stutsutttus) : : é : (3) 
Hence we find ae d e 
V=stut+stut+tsut+ust+s itu : s : (4) 
and are led to the algebraic system of equations 
stutstu Pa stu+tsv stutust 
Pp qd tis r 
stu ee ee ee 
= y% — Stutstut+tsutust+stu . - : (5) 
These equations may be simplified by dividing every term by s ¢v, and then as- 
suming 
t 
SE oe a 7 mee a: (Gl 
i 8 t uv 
They thus give 
styt+s _ stv+t _ stv'ty 
i 2 ff v 
= = stv'ts' +t t+u+l j ‘ : 5 (7) 
