172 Prof. Boole on certain Propositions in Algebra 



tions we may add the inequations 



\ = 0, p = 0, v = 0, p = 0, cr = 0. . . (17) 



The mode of eliminating \ fju, v, &c. from the above combined 

 systems is given in my paper ' On the Conditions by which the 

 Solutions of Questions in the Theory of Probabilities are limited/ 

 published in this Journal (August 1854). And the actual eli- 

 mination leads to the following results : viz. 1st, that p, q, and r 

 must be positive fractional quantities; 2nd, that they must 

 satisfy the conditions 



q+r>p, p+ r >q> p+i> r - 



These conditions, then, at least must be satisfied in order that 

 the system (15) may admit of solution in positive values of x, y, z. 

 To the quantity r they assign the inferior limits^ — q and q— p, 

 and the superior limits p + q and 1 . 



Exactly the same method is applicable to every form and 

 variety of the system (14), the final result always being a system 

 of linear inequations connecting the fractional positive quanti- 

 ties p, q, . . r. 



The conditions above determined are, moreover, the only ones 

 which are necessary in order that the system (14) may admit of 

 a solution in positive finite values of the variables x, y, . . z, that 

 case alone excepted in which one of the variables appears as a 

 common factor in, or is wholly absent from, the terms of V, 

 such variable admitting then the values and cc . This pro- 

 position is true in the case of n=l, as is obvious; and it may be 

 shown that if it is true for any particular value of n, it is true 

 for the next greater value, and so on ad infinitum. Whence its 

 general truth follows. The most important steps of the proof I 

 subjoin. 



Assigning to z in the n — \ first equations of the system (14) 

 any particular value, those equations degenerate into a system 

 of equations among the n — 1 variables x, y, &c., from which, by 

 assumption, a single positive set of values of those n — 1 variables 

 may be determined. These values substituted, together with 

 the assumed value of z in the first member of the last equation 



V 

 of the system —, will give to that member a corresponding 



value t. In order, then, that the system (14) may admit 

 of a solution in positive values of x, y, z, it suffices that the 

 limits of the quantity t, as z varies through the whole extent of 

 positive magnitude from to oc, should not transcend those 

 of r. For if the limits of r be wider than those of /, then r will 

 admit of values, which, while they satisfy the linear conditions 



