474 Mr. H. Wilbraham on the Theory of Chances 



Aj, A^, and the other two represented by s and / (from which 

 assumed independence the two equations are derived), is either a 

 datum of the problem or a condition necessarily recognized by 

 the mind ; the absurdity of this is shown enough by the fact, 

 that the latter two are purely imaginary events. 



Mr. Cayley's solution is, in fact, as follows : he introduces the 

 subsidiary quantities \, X^ determined by the equations 

 p^ = \+ {l—\)\^c<2 



and finds u by the equation 



In the preceding notation, and according to the meaning which 

 Mr. Cayley attaches to the subsidiary quantities \\c^ we have 



r = c,C2(l-X,)(l-A2) 



V = C2(1-C,)(1-X2) 

 t=Ci(l-C2)X, 

 r' = Ci(l-C2)(l-Xi) 



(T' = {\-Cy){l-c^); 



values which, combined with the equations for the determination 

 of \, \, satisfy, as they should do, the fundamental system of 

 relations between |, ^', ■??, rj', f, f', a'. But the equations last 

 written down give also 



or, as they may also be written, 



'n + rf a' ' 



i. e. 



Prob. Aj, A2, not E _ Prob. A,, not A^, not E 

 Prob. not A„A2,not E ~" Prob. not Aj, not A2, not E 

 and 



Prob. A,, A2 _ Prob. A), not A g 

 Prob. not Ap Ag ~" Prob. not A„ not Ag ' 



which are the assumptions made in Mr. Cayley's solution ; it is 

 clear that they amount to this, viz. that the events Aj, Ag are 



