Mr. A. Cayley on a Question in the Theory of Probabilities. 471 



tion. Using, as before, ABE, A'BE, &c. to denote the probabi- 

 lities of the compound events ABE, A'BE, &c, Mr. Wilbraham 

 in effect shows that in each of the two solutions the following 

 equations are (as they obviously should be) satisfied, viz. 



ABE + ABE' + AB'E + AB'E' + A'BE + A'BE' + A'B'E' = 1, ^ 



ABE + ABE' + AB'E + AB'E' = a , 



ABE + ABE' .... + A'BE+A'BE' . . =& U a ) 



ABE . . +AB'E =op, J 



ABE ....... + A'BE . ... . =j3q;J 



but (besides these) that, on the one hand, Prof. Boole has the 

 relations 



ABE _ AB'E ABE' _ AB'E' 



A'BE "A'B'E" A'BE' ~ A'B'E" ' * * W 



which equations are consequently implicit assumptions in his 

 theory, and which, with the equations (a), give his solution, 

 and that, on the other hand, I have the relations 



• ABE + ABE' _ AB'E 4- AB'E' ABE' _ AB'E' 



A'BE + A'BE' ~ A'B'E + A'B'E" A'BE' T A'B'E" ' W 



which are consequently implicit assumptions of mine, and which, 

 with the equations («), lead to my solution, — the signification of 

 these two equations being that the events A, B are treated as in- 

 dependent (1) in the case where it is not observed whether E 

 does or does not happen, (2) in the case where E does not 

 happen. 



The second of the equations (b) is the same as the second of 

 the equations (c) . But it is not easy to explain the first of the 

 equations (b) ; indeed Mr. Wilbraham remarked that it appeared 

 to him not only arbitrary but eminently anomalous. The pecu- 

 liarity in its form is, that it does not, like the others, when ABE, 

 &c. are considered as products, reduce itself to an identity ; it 

 seems to be a conclusion which, in support of his theory, Prof. 

 Boole is bound to justify a posteriori. 



Prof. lAlc wishes me to mention that he has succeeded in 

 obtaining a demonstration of the analytical theorem arising from 

 his theory, referred to in his " Reply " in my paper. 



2 Stone Buildings, W.C., 

 May 7, 1862. 



