connected with the Theory of Probabilities. 167 



pies above stated, from the events whose probabilities are given, 

 and whose logical connexion is explicitly determined from their 

 definition, to events which are free, and from which by limita- 

 tion the former are derived. Of those "free" events, the unknown 

 probabilities s, t, &c. are involved in the final algebraic equa- 

 tions to which the method conducts. As probabilities they ought 

 to admit of positive fractional values. Now it will be shown 

 that they do admit of such values when the data represent a 

 possible experience, and not otherwise. The conditions of pos- 

 sibility in the data are identically the conditions of mathematical 

 consistency in the method. Other important consequences are 

 connected with this demonstration, to which attention will be 

 directed. 



Proposition I. 



Let V be a rational and integral function of the n unknown 

 quantities x, y, ... z, consisting wholly of positive terms, and in- 

 volving all such terms as can be formed ivithout introducing powers 

 of x, y, ... z higher than the first ; also let V represent the sum of 

 those terms in V which contain x as a factor, Y p the sum of those 

 which contain y as a factor, and so on. Then the system of equations 



p, q, ..r being positive fractions, admits of one solution, and of only 

 one, in positive values of the quantities x, y, . . z. 

 To exemplify this proposition, let us suppose 



V = axyz + byz -+- cxz + dxy + ex +fy +gz + h, 

 then it is affirmed that the system of equations 



axyz + cxz + dxy + ex 



axyz + byz + cxz + dxy + ex-\-fy+gz + h 



(2) 



axyz + byz + dxy +fy ,„. 



axyz + byz + cxz + dxy + ex +fy +gz + h 



axyz + byz + cx z +gz _ ? , 



axyz + byz + cxz + dxy + ex +fy -tgz + h 



all the coefficients a, b, c, &c. being positive and the quantities 

 p, q, r positive and fractional, admits of one, and only one, solu- 

 tion in positive values of x, y, z. 



It is evident that the proposition is true when n—\; for 

 then we have Y = ax+b, and the system (1) is reduced to the 

 single equation 



ax 

 ax + b ~ P ' 



