594 
NOTES AND REFERENCES. 
119. I attach some value to the process here explained: the most simple 
application is that referred to at the end of the paper, for the factorial binomial 
theorem; to multiply m + n by m + n — 1, we multiply the m by (m — l) + n, and the 
n by m + (n — 1), thus obtaining the result in the form m (m — 1) + 2mn + n(n — 1), and 
so in other cases. 
121. The papers and works relating to the Question are 
1. Boole. Proposed Question in the Theory of Probabilities, Camb. and Dubl. 
Math. Jour. t. VI. (1851), p. 286. 
2. Cayley. 121, Note on a Question in the Theory of Probabilities, Phil. Mag. 
t. VI. (1853), p. 259. 
3. Boole. Solution of a Question in the Theory of Probabilities, Phil. Mag. 
t. vii. (1854), pp. 29—32. 
4. Boole. An Investigation of the Laws of Thought, on which are founded the 
Mathematical Theories of Logic and Probabilities, 8vo. London and Cambridge, 1854 
(see in particular pp. 321—326). 
5. Wilbraham. On the Theory of Chances developed in Prof. Boole’s Laws of 
Thought, Phil. Mag. t. vn. (1854), pp. 465—476. 
6. Dedekind. Bemerkungen zu einer Aufgabe der Wahrscheinlichkeitsrechnung, 
Crelle, t. L. (1855), pp. 268—271; 
viz. Boole proposed the question in 1, I gave my solution in 2, Boole objected to it 
in 3, and gave without explanation or demonstration his solution, referring to his then 
forthcoming work 4, which contains (pp. 321—326) his investigation. Wilbraham in 5 
defended my solution, and criticised Boole’s: and finally Dedekind in 6 (which does 
not refer to 4 or 5) completed my solution, by determining the sign of a radical, 
and establishing between the data, as conditions of a possible experience, the relations 
p — /3q and q —op neither of them negative. 
I remark that although Boole in 1, 3, and 4 speaks throughout of “causes,” 
yet it would seem that he rather means “concomitant events”: I think that in his 
point of view the more accurate enunciation of the question would be—The probabilities 
of two events A and B are a and /3 respectively; the probability that if the event 
A present itself the event E will accompany it is p, and the probability that if the 
event B present itself the event E will accompany it is q; moreover it is assumed 
that the event E cannot appear in the absence of both the events A and B: 
required the probability of the event E. 
He makes no assumption as to the independence inter se of A, and B: and 
moreover, in thus regarding A and B as events instead of causes, there is no room 
for regarding E as a consequence of one or the other of A and B, or of both of them. 
In my solution I regard A and B as causes: I assume that they are independent 
causes; and further that either or both of them may act efficiently so as to