CHAP. XIX.] OF STATISTICAL CONDITIONS. 311 



question will be the highest minor limit of w. To the above we 

 may add the relations similarly formed for the determination of 

 the relations among the given constants ft (s), n (/) . . n (1). 



14. The following somewhat complicated example will show 

 how the limitation of a solution is effected, when the problem 

 involves an arbitrary element, constituting it the representative 

 of a system of problems agreeing in their data, but unlimited in 

 their quaesita. 



PROBLEM. Of n events x l x z . . x n , the following particulars 

 are known : 



1st. The probability that either the event x l will occur, or 

 all the events fail, is pi . 



2nd. The probability that either the event x 2 will occur, or 

 all the events fail, is p 2 . And so on for the others. 



It is required to find the probability of any single event, or 

 combination of events, represented by the general functional form 

 (x l . . x n ), or 0. 



Adopting a previous notation, the data of the problem are 



Prob. (Xi + x l . .x n )=pi . . Prob. (x n + x t . . x n ) = p n . 



And Prob. (x l . . x n ) is required. 

 Assume generally 



X r + X V . . X n = S rt (1) 



t = w. (2) 



We hence obtain the collective logical equation of the problem 



2 {(x r + x l . . x n ) s r + s r (x r - #1 . . Xn)} + $w + w!j> = 0. (3) 



From this equation we must eliminate the symbols x l , . . x n , and 

 determine w as a developed logical function of s l . . s n . 



Let us represent the result of the aforesaid elimination in the 

 form 



Ew 



then will E be the result of the elimination of the same symbols 

 from the equation 



2 {(X r + X l .. X n ) S r + S r (x r - X l . . X n )} + 1-0 = 0. (4) 



Now E will be the product of the coefficients of all the con- 

 stituents (considered with reference to the symbols x l , x. 2 . . x n ) 



