OP ARTS AND SCIENCES : MARCH 12, 1867. 255 



Given <p x =F 1- 



Required to eliminate x. 

 Let q>' x == 1 — <jp x = 



9>' (1) ,9' (0) =5= (1 - 9 (1)) > (1 - V (0)) =F 

 l-(l-g>(l)), (1-9,(0)) =1. 



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

 instead of to x and y, 



1 - (1 - g, (1)) , (1 - 9 (0)) = qp (1) , «P (0) + » (1) , (1 — <p (0)) 



+ 9(00,(1 — 9(1)). 



But by (18) we have also, 



9(l)+9(0)=F9(l),9(0) + 9(l),(l-9(0))+9(0),(l-9(l))- 



So that 



(26.) 9 (1) + 9 (0) =7= 1 when q> x == 1. 



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 = b a = b 



(28.) a + b = (a -\r b) + (a , b). 



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

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

 happens. 



(29.) ab a = a ,b. 



It will be convenient to set down some obvious and fundamental 

 properties of the function b a . 



(30.) ab a = ba b 



(31.) cp (b a and c a ) = (cp (b and c)) a 



(32.) (1 — ■ b) m = 1 — b & 



