ON HAMILTON’S NUMBERS. 
307 
or, since fx q* — f q is a positive quantity, 
P>{q- Vqf, 
at all events when q = or > 462. 
It will be found also on trial that this formula remains true for all the values of q 
inferior to 462. 
Thus 
462 > (24 — v /24) 3 , 
24 > ( 6 - x/o) 3 , 
6 > ( 3 - V3f, 
3 > ( 2 - y/2 ) 3 . 
Hence, universally, 
v>(q — Vqf* 
But we know that 
We may therefore write 
p < q 2 . 
p=(q — 6V qf, 
where 6 is some quantity between 0 and 1 . 
Similarly, 
q — (r — 0 1 v 7 rf, 
r = (s — 0. 2 \/ s) 2 , 
where 0 V 6 2 , ... are also positive fractions. 
When p and q become infinite, 
Hence the ultimate value of 6 is Similarly, 0 l} 0 2 , ... all of them converge to 
the value 3 -. 
This agrees with the result previously demonstrated (p. 295), and is the starting 
point of all that follows. 
We know that letters p, q, r, s, . . ., being used to denote the halves of the 
augmented Hamiltonian Numbers, they are connected by the scale of relation 
P = 2 + 
q (2 q - 1) r (2 r - 1) (2 r - 2) 
2 ~ 2.3 
+ S-T, 
* Had this inequality been true only for values of q sufficiently great, it would have been enough for 
the purposes of the text. 
