650 PROFESSOR BOOLE ON THE COMBINATION 



it is evident that if s, t, and v are positive quantities, and if we write 



stv s .. t V 



T =u, ^ = \ v =fx, — = v 



u, A, fx, and v must be positive fractions, whence, in addition to the equations 



u + A = p 



u + fJL — q 



u + v = r 

 we shall have the inequations 



«>0 X>0 A«>0 t/>0 



u + \ + fx + v< 1 



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, t, &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 true. I will here only exemplify it in Problem II. , 

 Art. 34. 



Here the value of V is 



V = xyst + xs + yt + tcy + cc + y + l 



and the algebraic system employed in the determination of Prob. xyz is 



xyst + xs + xy + oc _ xyst + yt + xy + y 

 c c' 



_ xyst + xs _ xyst + yt _ xyst 

 cp c' q u 



= xyst + xs+yt + xy + x + y + l ' . . . (16) 



For the determination of u we hence find the following equation — 



u(u — c'+cp — 1) (u — c + c'q — 1) — (cp — u) (c'q — u) (u — cp + c'q — l)=.0 (17) 



