ON HAMILTON’S NUMBERS. 
293 
it will be true universally; for in all the succeeding series the term we have called R 
will be higher than the term G in the scale 3, 4, 6, 12, 48, . . . 
Hence 
P — 1 = or < \ (Q 3 — Q). 
For the initial values of Q, P, (viz., 3, 4,) 
p_ 1 == l(QS_Q). 
[When P represents any term beyond the first it is very easy to prove, but too 
tedious to set out the proof, that the sum of all the terms after the first in the 
series equated to P — 1 will be less than — 2 ; so that, except in the case stated, 
P < £ (Q 3 - Q)]. 
For the series 12, 48, 924, ... we have seen that P > 9Q 3 /32. 
Hence, for the series 48, 924, . . . , 
But 
Hence 
Q 2 - Q R(R- 3)(R-2) 
y> 2 6 ’ 
Q 8 -Q _ E? 
> 2 6 ' 
p> 
Q 2 — Q 64 
8 1 
Q 1 , and P < 
Q 2 — Q 
Hence, when P, Q, are at an infinite distance from the origin, 
Hence, also, 
(fflr uItimatel y = I = *’ 
which proves the theorem left over for “ exact proof” in the memoir referred to. 
It is convenient to deal with the halves of the sharpened Numbers of Hamilton, 
which may be called the reduced Hamiltonian Numbers, and denoted by h with a 
subscript, or, when required, by p, q, r, . . . (the halves of P, Q, R, . . . respectively). 
We have then 
2 P< 
4 if - 2 q 
* Numbers increased by unity may conveniently be denominated sharpened numbers, and numbers 
diminisbed by unity flattened numbers. 
