ON HAMILTON’S NUMBERS. 
295 
and we have previously shown that 
P>T~ ¥r T ~ 2’ 
at all events when P is not lower in the scale than E G . 
The fraction arises from our having substituted for II 3 the inferior value 
(- 9 - Q) f ; but, the higher we advance P in the scale, the nearer R approaches to 2Q, 
and is ultimately in a ratio of equality with it. But, if we had written (2Q)= for R 3 , 
the coefficient, which now stands at — -g^-, would have been — f. In like manner, 
as P and Q are travelled on in the scale, R 3 and S 4 become indefinitely near to 2Q 
and (2R) 3 , i.e., 8Q, so that the coefficient of Q in the superior limit approximates 
indefinitely near to 
~ i + I + i- e -> fj 
and the two limits of p which have been obtained become 
+ (|+ e)q, 
c f - (I + v) r - i ( b 
where ultimately e and q are infinitesimals.* 
Hence it follows that the ultimate value of 
(p - 2 3 ) -5- r is - b 
i.e., 
2E ; - E**_, . 
—-= — a/a when i = co. 
. v 9 
l -1 
Let X, /A, v, . . . represent the halves of .the Hypothenusal Numbers in the triangle 
given at the commencement of the paper, i.e., the differences of the numbers which 
we have called p, q, r, . . . 
Since 
p — cp — and q — r 2 — § r\ 
p — q = <f — ■§■ q* — q, and q — r — r 2 — § r 1 — r. 
Obviously, therefore, as a first approximation when X, p, are very advanced terms 
in the hypothenuse, 
X = p 2 . 
Let us write 
X = p. 3 -\- Kp a 
for a second approximation. 
# As a matter of fact, it will be found that, as soon as q and p attain the values 6, 24, cp •— -f <p may 
be taken as a superior limit. It may be noticed also, to prevent a wrong inference being drawn from the 
above expressions, that, as will hereafter appear, rj is an infinitesimal of the order 1 /q*, when q is 
infinite. 
