OF TESTIMONIES OR JUDGMENTS. 617 



of the events x and y is known. Symbolically expressed, it is the value of the 

 fraction. 



Prob. x y z 

 Prob. x y 



or, as it may, by resolving the denominator, be written, 



Prob. x y z 

 Prob. x y z + Prob. x y z ^ ' 



To effect this object, we shall determine the value of Prob. xy % and Prob. xy~z 

 separately. 



Collecting the elements furnished by the preceding analysis, the first of the 

 partial problems herein involved may be thus stated : — 



Given Prob. x=a 1 Prob. y=a 2 ~j 



Prob. w = a l c 1 Prob. v=a 2 c 2 \ ... (11) 



Prob. wz=a 1 c 1 p 1 Prob. vz=a 2 c 2 p 2 J 



with the conditions, wx=0, vy=0, wv=0, . . . • (12) 



Required u, the value of Prob. xyz. 



In selecting the above, I have chosen to employ (3) in place of (1), and (6) in 

 place of (^4). It makes no difference in the final result. 



In accordance with the rule, let us write 



wz=s, vz=t, xyz=(p . (13) 



we must then from (12) and (13) determine as a developed logical function of x, 

 y, w, v, s, and t. 



This problem admits of perfectly definite solution on the principles of the cal- 

 culus of Logic. I shall here merely give the result, and point out a method by 

 which it may be independently verified. We find 



(f)=xyiusvt + xyvtws+Q(xwsyvt -Yyvtxws +xy wvst 



+ y x w v s t + xy w v s t + xy w v s t -Yxwy v s t + xyv w s t + yvxws t) 

 + ^xywvli 1 + terms whose coefficient is ^ (14) 



We may verify this expansion by substituting for s and t their values wz and 

 tz, paying attention to the conditions (12), and then comparing the result with 

 the value of <£, viz., xyz. 



Thus the term xywsvt becomes, on substitution 



xy w v xw z x (l—vz)=xyzwv 



by the calculus of Logic. Now this represents a class entirely included in the 

 class xyz, whence the coefficient of the term is unity. 



The term xwsyvl reduces to x w z y u, and represents a class no part of which 

 is included in xyz, whence the coefficient is 0. 



VOL. XXI. PART IV. 8 D 



