OF TESTIMONIES OR JUDGMENTS. 



633 



we find, by the calculus of logic, . 



<p =xy st + (xsy t +ytx s +xy s t +xy s t +yx s t+xyst) 



1 



+ terms whose coeficients are 







(4) 



a result which may be verified by the method applied to (14) in Art. 23. 

 Hence we find, adopting the simplification of Art. 21 , 



Y=xyst + xs + yt + xy + x + y + 1, 

 and since we have 



Prob. x=c, Prob. y — c', Prob. s=cp, Prob. t — c'q | 

 Prob. xyz = u, J 



we find, as an algebraic system of equations, 



xyst + xs + xy + x xyst + yt + xy + y 



(5) 



\ 



_xyst + xs 

 cp 



xyst + yt 

 c'q 



(6) 



xyst , , n 



= — - — = xyst + xs + yt + xy + x + y+l 



This system is easily reduced to the form 



xs yt xy + x + y + 1 



cp — u c'q — u 1 + u—cp — c'q 



x + 1 y + 1 xsyt 



u 



(7) 



1 + u — cp-c' 1 + u — c — c'q 



And if we equate the respective products of the first three and of the last three 

 members of the above, we find 



(cp — u) (c'q — u) (1 + u—cp — c'q) = (l + u — c' — cp) (l + u — c—c'q)u . • (8) 



a quadratic equation by which the value of u must be determined. 

 If, in like manner, we assume 



Prob. xyz = t 

 we shall find 



(cl-p-t) (c'T^q-t)(l + t-cT^p-c 7 l^q) = (l + t-c'-cT^p) (l+t-c-c'l-q)t (9) 



From these equations the values of u and t being determined, we have finally 



Prob. xyz u 



Prob. xy u + t 



(10) 



Before we can apply this solution, we must determine the conditions of pos- 

 sible experience, and the conditions limiting the values of u and t. For this pur- 

 pose writing 



Prob. xyz—u, Prob. xyz—t, Prob. xzy — [i. Prob. xyz — v, 



Prob. yzx =p, Prob. yxz =■&, 



VOL. XXI. PART IV. 8 H 



