connected with the Theory of Probabilities. 177 



we obtain a system which agrees with (15), the constants a, b, 

 c, d being made severally equal to 1. 



The conditions which must be fulfilled in order that the trans- 

 formed system may have a single solution in positive values of 

 x, y, z, depend, as it has been seen, on the analysis of (16), (17). 

 Let us then seek what would be the conditions of possible expe- 

 rience in a problem, in the solution of which we should be con- 

 ducted by the general method in probabilities to the algebraic 

 system (24). 



That system must arise from a logical equation 



V = l, (25) 



and a connected system of data, 



Prob. s=p, Prob. t—q, &c, 



s, t, &c. being logical symbols. Moreover, V as a logical func- 

 tion is identical in form with V as an algebraic function (23). 

 Now by virtue of (25) all constituents vanish except those which 

 are found in V. Hence 



Prob. s= Prob. {stu + st(l— u) +s(l — t)u}=p\ , 9fiV 

 Yrob. t= Yvoh. {stu + st{l—u) + {I- s)tu}=q J' 



Moreover, as V = 1 denotes, as a logical equation, the certainty 

 of the event V, we shall have 



Prob. V = l (27) 



Hence if we make 



Prob. stu = X, Prob. (1 —s)tu = p,, Prob. s(l — t)u= v, 

 Troh.st{l-u)=p, Prob. {l-s){l-t){l-u)=a, 

 we shall have from (26) and (27) 

 X+v+p=p 

 \ + /j, + p = q 

 X + /j, + v — r 

 \ + fji + v + p + cr=l; 

 to which, X, p., v, &c. being probabilities, we may add 

 \ > 0, /u, r; 0, v if 0, p ^ 0, a :f. 0. 



This combined system is, moreover, identical with the com- 

 bined system (16), (17) ; and the mode of elimination being the 

 same, it will furnish the same final relations among p, q, r, for 

 the conditions of possible experience, as arc furnished by (16), 

 (17) for the conditions of algebraic solution of (15) in positive, 

 and as we have afterwards shown, of (21) in positive fractional 

 rain B of the variables involved. 



Phil. Mag. S. 4. Vol. 9. No. 58. March 1855. N 



