650 PROFESSOR BOOLE ON THE COMBINATION 
it is evident that if s, 4, and v are positive quantities, and if we write 
stu § t v 
7 oe ye Ny ee 
u, A, #, and vy must be positive fractions, whence, in addition to the equations 
Uut+N= Pp 
u+m=q 
utv=r 
we shall have the inequations 
u>0 AS0 MSO yS0 
u+A+p+vel 
This system is identical with the one obtained in 10, Art. 13, for the determi- 
nation of the conditions of possible experience in the particular question of Proba- 
bilities, in which the above function V presents itself. And a very little attention 
will show, that if in any case we express as above the relations which must 
obviously be fulfilled in order that s, ¢, &c., may be positive quantities, we shall 
form a system of equations and inequations precisely agreeing with those which 
we should have to form in order to obtain the conditions of possible experience, 
if we sought those conditions, not from the data in their original expression, but 
from the translated data, as employed in Art. 13. 
Hence, in order that s,t . . in the system of Art. 21, may be positive, or in the 
prior system, positive fractions, the problem of which these systems of equations in- 
volve the solution must represent a possible experience. 
Conversely if that problem represent a possible experience, the quantities s,t . . 
will admit of being determined in the system of Art. 21, in positive values, or in the 
prior system, in positive fractional values. 
I have not succeeded in obtaining a perfectly rigorous proof of the latter, or 
converse proposition in its general form, but I have not met with any individual 
cases in which it was not trne. I will here only exemplify it in Problem IL, 
Art. 34. 
Here the value of V is 
V=ayst+ast+ytt+ay+at+ytl 
and the algebraic system employed in the determination of Prob. zyz is 
ayst+ ast aryt+@2_ xyst+yt+ayt+y 
¢ i c 
_ ayst+as_ axystt+yt _ xyst 
een Te GMs Sy 
=aysttastytt+aytatyt+1 |... (16) 
For the determination of w we hence find the following equation— 
u(u—c¢+ep—1) (u—c+e' g—1)—(cep—u) (ce g—u) (w-cp+¢ q—1)=0 (17) 
