ON HAMILTON’S NUMBERS. 
287 
1786 (a matter of a couple of pages), Hamilton’s Report of 1836, with the tract of 
Mr. Jekrard therein referred to, and the memoir in ‘Crelle’ of December, 1886, 
constitute, as far as we are aware, the complete bibliography of the subject up to the 
present date. 
§ 1. On the Asymptotic Laws of the Numbers of Hamilton and their Differences. 
Consider the followings Table :— 
10 0 0 
0 
0 
0 
0 .... 
1 1 1 
1 
1 
1 
1 .... 
2 3 
4 
5 
6 
7 .... 
6 
15 
29 
49 
76 .... 
36 
210 
804 
2449 .... 
876 
24570 
401134 .... 
408696 
246382080 .... 
83762796636 .... 
Any line of figures, say p, 
q, r 
5 S, t 
... 6, in 
the Table being given, to form the 
ubsequent line q 1} r l3 s 1} t v . 
. 6 X , we 
write 
p(p + l) , 
^ = —2 —+ «• 
_ p(p + 1) (2p + 1) 
r^ 
1.2.3 
+ pq + r. 
o O + 1)Q + 2) (3 p + 1) p(p + 1) 
— - - - - "Tiot ~r P r “T s - 
1 . 2 . 3.4 
1.2 
p(p + l)(p + 2)(p + 3)(4p + 1) , p(p + l)(p + 2) _ .p(p + 1) . , , 
1.2.3.4.5 + 1.2.3 q + T.2 ' * + ^ + L 
q _ p(p+D..-(p + i-l)(ip + 1) , p (p - 
1 12.3...C + 1) ^ 1 
+ 1) . . . (p + i — 2) 
•3 ... (i — 1) 
q + . . . + 6. 
If we call the «, tb term of the m th line [to, n\ the general law of deduction may he 
expressed by the formula 
[m + 1 ,n\— — B„ +1 ([to, 1] — 1) + t [m, % -f 1 ~ %] B,-[m, l], 
i = 0 
where B, Jc means the coefficient of z l in (l — z) k . 
