developed in Professor Boole's " Laws of Thought." 469 



Out of the events represented by V there are two, ot and v, 

 which imply that A occurs but not B ; consequently the chance 

 of A occurring but not B, which is the required chance and may 



^n v)(\ s) 



be called u, = -^ K^ ' . From these five equations x,y,s,s 



may be eliminated, and there remains an equation which gives u. 

 Or the values of x, y, z, and s may be found from the first foui' 

 equations, and thence the value of any function of them is known. 

 Tliis method of solution is almost identical with Professor 

 Boole's. The assumptions are the same in both, and they differ 

 only in my examining as above the import of each step taken 

 separately. Representing the chances of the sixteen separate 

 compound events by the Greek letters prefixed to them, the 

 condition that the four events A, B, C, and S are mutually inde- 

 pendent is equivalent to the following relations among S, e, &c. 



S Q i K tB p T V 



8 e t K _ fJ, V T ^ 



o e 6 K X V p V 



i/jLi!rT-\lr(f)V(o 



o e ^_i ^ At CT ^fr 



K V p "^ X 4^ ^ ^ 

 These are reducible to eleven independent equations, viz. the 

 seven in the first hne and ;3 = -=- = -, and - =-. The state- 



U tS p V IT 



ment that the four simple events are independent is only a con- 

 cise way of stating that these eleven equations are assumed to 

 hold good. The assumption of these eleven is equivalent to 

 saying that 8, e, &c. are proportional to xj/zs, (1 —x)yzs, &c. 



We have taken S to represent a simple event of which the 

 absolute chance is s, not to represent the concurrence of A, B 

 and C ; and when eight out of the sixteen compound events were 

 struck out as implying the concurrence of the events which we 

 know to be incompatible, we did not make S identical with A, 

 B, and C concurring, but we only say that the cases in which S 

 is accompanied with the absence of A, B and C, or of any of them, 

 arc beyond our universe of observation. The truth of the eleven 

 assumed equations is sup])osed throughout the ])robleni ; for if 

 at any point we were to discard or deny them, all conclusions 

 based on them would fall to the ground. 



It may naturally be asked, how comes it that when there were 

 given only the chances of the events A, B and C, we assumed 



