92 Prof. Boole on the conditions by which the Solutions of 



chances in which the data are not the probabilities of simple in- 

 dependent events. I propose in this paper to develope an easy 

 and general method of determining such conditions. This object 

 has been attempted in Chapter XIX. of my treatise on the ^ Laws 

 of Thought.' But the method there developed is somewhat dif- 

 ficult of application, and I am not sure that it is equally general 

 with the one which I am now about to explain. I premise the 

 following proposition. 



Proposition, — To eliminate any symbol of quantity a? from any 

 system of inequations in the expression of which it is involved. 



The general method will be best explained by an example. 

 Suppose it required to eliminate x from the inequations 



3y—x—z^0 



x-2y-hz = 0. 



Reducing each of these inequations to a form in which the first 

 member shall be x, we have 



a^^z—y 



x^Sy—z 



x = 2y''Z. 



From these equations it appears that x has for a superior 

 limit Sy—z, and for inferior limits z—y and 2y—z. As the 

 superior limit must in general exceed each of the inferior limits, 

 we have 



Sy-z = z-y, 3y-z=2y-z, 

 whence 



And these are the only conditions which are independent of x. 



The general rule would therefore be to seek from the sevei-al 

 inequations the superior and inferior limits of x, and then to express 

 by new inequations the conditions that each superior limit shall be 

 equal to, or greater than, eveiy inferior limit. 



If it is a condition that a? is a positive quantity, then must 

 each superior limit be made J j or we might add to the system 

 of inequations the inequation x^O, and apply the general rule. 



When several quantities, as x, y, &c., are to be eliminated, we 

 can proceed by first eliminating x, then from all the inequations 

 which either result or remain eliminating y, and so on. - 



It is obvious that the number of inequations obtained by the 

 elimination of a symbol may greatly exceed that of the inequa- 

 tions from which the elimination has been effected. 



