Questions in the Theory of Probabilities are limited, 97 



in what cases the problem becomes determinate in the sense above 

 explained. Thus in the particular problem discussed in Pro- 

 position 2, since we have 



it follows that whenever one of the upper limits of w becomes 

 equal to one of the lower_, the other conditions remaining satisfied, 

 the problem becomes determinate. Thus^ if we have 7?2 = 0, we 

 find from the above^ 



Now as lu cannot at the same time be both greater and less than 

 Cip^f it must be equal to c^p^; the other conditions simply redu- 

 cing to 1 — ^2 > ^\Pv ^^^ solution, therefore, is 



w=zc^p^, 

 the data being necessarily connected by the condition 



Let us apply to this case the solutions of the general question 

 in probabilities respectively given by Mr. Cayley and myself. 

 Mr. Cayley's solution is expressed by the quadratic equation 

 (\-c,(l-p,)-w')(\-c^{\-p^)-w) = {\-c,)(\-c^)[\-w).{n) 

 If we make jt?2=0, it becomes 



(1 -Cj(l -p^) -m;)(1 -Cg-w) = (1 -cy) (1 -Cg) (1 -m;), 



and this equation is not satisfied when we make w=:c^Py. The 

 solution which I have given is contained in the quadratic equation 



{w-^c^p^)[w-Cc^p^)[l-w) = (\-c^{\-p^)-w) 



(}-Cc,[l-p^-w)[c,p^-\-Cc,pc,-w)', . . (12) 



and this equation, on making jOg— ^^ i^ satisfied by the value 

 w=-c^Pi. The reader may examine for himself, and with exactly 

 similar results, the class of cases in which the data happen to be 

 connected by the relation 



CiPx + c^{l-p^=:\, 

 or by the relation 



C2^2 + c,(l-i^i) = l. 



But there is another and more remarkable distinction to which 

 I would advert. I have shown (Laws of Thought, p. 324), that 

 in all cases in which the data of the abx)ve general problem are , 

 possible, the quadratic equation (12) furnishes one root, and only 

 one, falling within the limits assigned by the method of this 

 chapter. It is needless to remark that Mr. Cayley's equation 



Phil. Mag, S. 4. Vol. 8. No. 50. Aug, 1854. H 



