306 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



with A in the expression of w, a combination which, as shown in 

 various examples of the Logic we are permitted to make, than 

 we other wise should obtain. 



Finally, as the concluding term of the development of w in- 

 dicates the equation D = 0, it is evident that n (Z>) = 0. Hence 



we have 



Minor limit of n (Z>) < 0, 



and this equation, treated by Prop. 3, gives the requisite condi- 

 tions among the numerical elements n(s), n(t), &c., in order that 

 the problem may be real, and may embody in its data the re- 

 sults of a possible experience. 



Thus from the term - (1 - s) t in the second member of (2) 



we should deduce 



n(\ -s) + n(t) -n(l)<0, 



These conclusions may be embodied in the following rule : 



10. RULE. Determine the expression of the class w as a deve- 

 loped logical function of the symbols s, t, fyc. in the form 



Then will 



Maj. lim. w = Maj. lim. A + C. 



Min. lim. w = Min. lim. A + D. 

 The necessary numerical conditions among the data being given by 



the inequality 



Min. lim.Z><rc(l). 



To apply the above method to the limitation of the solutions 

 of questions in probabilities, it is only necessary to replace in 

 each of the formulae n (x) by Prob. #, n (y) by Prob. y t &c., and, 

 finally, n (1) by 1. The application being, however, of great im- 

 portance, it may be desirable to exhibit in the form of a rule 

 the chief results of transformation. 



11. Given the probabilities of any events s, t, &c., whereof 

 another event w is a developed logical function, in the form 



