connected with the Theory of Probabilities. 175 



of the system (15), the first member of the final equation will, 

 when z = 0, coincide with the highest of the inferior limits of r 

 above determined, and when z = <x with the lowest of the superior 

 limits; for the highest inferior and the lowest superior limit 

 are the true limits of r, as above determined, on the assumption 

 that x, y, z are positive quantities. And therefore they are the 

 true limits of the first member of the final equations under the 

 same conditions. 



Thus it has been shown that as z varies from to infinity, 

 while x and y are positive quantities constantly determined so 

 as to satisfy the two first equations of the system (15), the first 

 member of the final equation of (15) will vary from the highest 

 of the inferior to the lowest of the superior limits of r. In the 

 variation it must once, or more often, coincide with the actual 

 value of r. But that it cannot more often than once coincide with 

 r is evident from the constitution of H„ (Prop. I.), which does not 

 permit that the function in question should ever resume a former 

 value while z continues to increase. For if such resumption 

 were possible, a maximum or minimum value of the function 

 would intervene ; and this, as H„ continues to consist of positive 

 terms only, cannot happen. One solution therefore exists, and 

 one only, of the system (15) in positive values of x, y, . . z when 

 the linear conditions of inequality among the quantities p,q, r 

 are satisfied, and then only. 



The reasoning is general, and serves to establish the general 

 truth of the proposition. 



Proposition III. 



The conditions which must be satisfied in order that the final 

 algebraic system of equations 



7"»— * (a2) 



to which, by a general method, the solution of questions in the 

 theory of probabilities is reduced (Laws of Thought, p. 270, 

 Phil. Mag. December 1854), admits of a single solution or posi- 

 tive fractional values ofs, t, fyc, are identical with the conditions 

 of possible experience in the data of the problem. 



The function V in this proposition is the sum of a series of 

 terms which we may by analogy term algebraic constituents. If 

 the number of variables s, t, &c. be two, the possible algebraic 

 constituents are 



st, s(l-l), {1-8% (l- s ){l-t), 

 and V consists of some or the whole ofkhese. V, is the sum of 

 those constituents of which s is a factor, and so on for the others. 

 The algebraic system (22) -is derived from a logical equation in 



