182 [June 19, 



a system of inferior limits expressed by linear functions of the other 

 quantities, and a system of superior limits also so expressed. 



Thus, if A, B, C represent any simple events, and if p l represent 

 the probability of the concurrence of B and C, p z that of the concur- 

 rence of C and A, p 3 that of the concurrence of A and B, then p l3 p 2 , p 3 

 must, in order that they may be derived from experience, satisfy the 

 conditions 



as well as the conditions implied in their being positive proper frac- 

 tions. 



On the other hand, the ideal events being by hypothesis simple and 



independent, the auxiliary quantities which represent their proba- 



bilities will be subject to no other condition a priori than that of 



being positive proper fractions to no other condition a priori, 



because their actual values are determined in the process of solution. 



Now the most general results of the analytical investigation are 



1st. That the auxiliary quantities representing the probabilities of 



the ideal events admit of determination as positive proper fractions, 



and, further, of a single definite determination as such, precisely when 



the original data supply the conditions of a possible experience. 



2ndly. That as a consequence of this the probability sought will 

 always lie within such limits as it would have had if determined by 

 actual observation from the same experience as the data. 



The proof of these propositions rests upon certain general theorems 

 relating to the solution of a class of simultaneous algebraic equations, 

 and, auxiliary to this, to the properties of a functional determinant. 

 The following are the principal of those theorems : 

 1st. If the elements of any symmetrical determinant are all of 

 them linear homogeneous functions of certain quantities a v 2 , . . . a r 

 if the coefficients of these quantities in the terms on the principal 

 diagonal of the determinant are all positive and if, lastly, the coeffi- 

 cients of any of these quantities in any row of elements are propor- 

 tional to the corresponding coefficients of the same quantity in any 

 other row, then the determinant developed as a rational and integral 

 function of the quantities a,, 2 , . . . a r will consist wholly of positive 

 terms. 



A nd, as a deduction from the above, 



