ON HAMILTON’S NUMBERS. 
305 
It is interesting to notice that the formula apparently remains arithmetically true 
for finite values of p and q, provided that q is not less than 24, when we replace each 
Hence, from what lias been shown above, 
= -§■ ( \/ 2 + */ r + \/ s + ... + V 6 ) — 2 p x. 
g 3 —p _ 
In this equation we may write 
•J r — qi + 
^ 1 ~~ (* — 3 equations), 
v t —■ gww ~f" ^3 
where 7^, 7c 3 , 7r 3 , . . . are all of them finite (and, as a matter of fact, of no consequence for our immediate 
object, positive proper fractions). For, ultimately, 
\ — V r — qi = 
r — q 3 
+ gi 
+? 
i = ¥\ = \ ( see p- 307 )> 
and consequently the finiteness of each 7; is a direct inference from the general principle previously 
applied in the case of the e’s. 
Applying this result to the equation previously given, it follows that gi + q* + ... + g®*~ 2 = -| A — vx 
(where v is finite) = F(g) + (g~s + g - * + ... + q~ ®* _s ) — { (z — z -1 ) + (zi — z~i) + (z* — z~i) + ...}’ 
where z lies between 1 and 2. 
The series of negative powers of q is obviously less than x, and the z-series, which follows it, is less than 
the finite quantity 2 (z — 1/z), i.e., < 2 (2 — ^). Hence §- A — F(g) + Ox, where 0 is a number lying 
between fixed limits, and x, the rank of q, is of the same order of magnitude as log log q. This equation 
contains as a consequence the asymptotic theorem to be proved ; for, using i to denote any positive integer, 
| A — Si q {i)i = F (g) - Si q {i - )l -Ox=q <^ +1 + ‘ s” ( q^ - S'o 1 lq^ +1 -Ox. 
s = i + 2 
Hence, remembering that x is of the same order of magnitude as log log q, and that 
S — 00 « 8 : _i_ o ? + 0 
S (g (i! -g- (i) ) < 2(g (i >‘ “-g - 1 * 1 ), 
s = i + 2 
which is of a lower order of magnitude than g (i) * + 1 , it follows that f A — Si q {i) ' for all values of i is ulti- 
mately in a ratio of equality with g (J) , which is the theorem to be proved. 
We have thought it desirable to obtain the formula ■§ A = Fg + 9x for its own sake, but, so far as 
regards the proof in question, that might be obtained more expeditiously from the expression given for 
3A/2 — vx without introducing the series Fg. 
It is easy to ascertain the ultimate value to which 0 converges. In the first place, the series of 
fractions 1 /gi + 1 /gi -f 1 /gw + ... to x — 2 terms (where x is the rank of g) may be shown to be 
always finite, and consequently, when divided by x, converges to zero. 
For we know that (p — g) > (g — r ) 2 > (r — s ) 4 . . . > (6 — 3) 2 * -2 . Hence the last term of the 
series gi, gi, gw . . . (viz., g®‘ _s ) > 3. Hence the finite seines 1 /gw + 1/gi + 1/gi + . . . for a double 
a fortiori reason is less than the infinite geometrical series v + v + + ... < \. 
[In fact, from § 1 (p. 299) it may easily be shown that the last term of the series gi, q*, qi . . . > M 4 
> (1-465)* > 4-608, so that the sum is really less than —V- .] 
Hence, retracing the steps by which 0 has been obtained, and observing that p' differs from p by a 
finite multiple of 1 jx, we have ultimately 0 = v = h — 3// = h — 3p — lc — 3e = ~ — j — — If, then 
(using u x to denote the half of the sharpened X th Hamiltonian number), we write u x — \ju x — v x , and 
understand by G (t — ljt) the infinite series (7i — t~i) + (ti — t~ 1) + (ti — t~ w) + . . ., it is easily seen 
that the principal part of -f(v x +f), regarded as a function of v x and x , is v x — 
MDCCCLXXXVII. — A. 2 R 
