OF ARTS AND SCIENCES: MARCH 12, 1867. 255 



Given qp a: = 1. 



Required to eliminate x. 

 Let q,' X ==\ — (fx=.0 



9' (1) ,<y' (0) -F (1 - <? (1)) , (1 - 9 (0)) - 

 1_(1_^(1)),(1_^(0)) =1. 



Now, developing as in (18), only in reference to 9) (1) and qp (0) 

 instead of to x and y, 



1 - (1 ~ q, (1)) , (1 - 9, (0)) = 9, (1) , g, (0) + 9, (1) , (1 - cy (0)) 



+ 9(0), (1-9(1)). 

 But by (18) we have also, 



9(l) + 9(0)=^9(l),9(0) + 9(l),(l-9(0)) + 9(0),(l-9(l))- 



So that 



(2G.) q> (1) + 9 (0) = 1 when (fx=l. 



Boole gives (25), but not (26). 



We pass now from the consideration of identities to that of equa- 

 tions. 



Let every expression for a class have a second meaning, which is 

 its meaning in an equation. Namely, let it denote the proportion of 

 individuals of that class to be found among all the individuals ex- 

 amined in the long run. 



Then we have 

 (27.) If a = & a = h 



(28.) a -\- h = {a -\r h) -\- {a ,b). 



Let b^ denote the frequency of ^'s among the a's. Then considered 

 as a class, if a and h are events bg_ denotes the fact that if a happens b 

 happens. 



(29.) ab^ = a ,b. 



It will be convenient to set down some obvious and fundamental 

 properties of the function b^. 



(30.) ab^ = ba^ 



(31.) (f (5a and c„) = (qp {b and c)), 



(32.) (1 _ i)^ = 1 _ h. 



^a 



