ON HAMILTON’S NUMBERS. 
291 
sum may be taken as a vertical sum of line-sums or as a horizontal sum of column- 
sums, and, although for licit values of t each sum has a finite value, the two finite 
values are not identical, just as a double definite integral may undergo a change of 
value when the order of its integrations is reversed.* 
I noticed at p. 478 of the 100 th volume of ‘ Crelle’ that the value of any Hamiltonian 
Difference divided by the square of the preceding one was always greater than and 
stated as morally certain, but “ awaiting exact proof,” that this ratio ultimately becomes 
By aid of Mr. Hammond’s formula for the numbers, I shall now be able to supply 
tlris proof, and at the same time to show that the ratio of a Hamiltonian Number to 
the square of its antecedent (which, of course, converges to the same asymptotic 
value 15 ) is always less than that limit.t 
We must in the first place prove that in the series 
— /3 3 Ei - 2 + /3j(E/_3 — /3 5 E ;_ 4 + . . . 
the absolute value of each term is greater than that of the one which follows it. 
In proving this, I shall avail myself of the property of the Hypothenusal Numbers 
disclosed in the process of forming the triangle given at the outset of the memoir, viz., 
that E,- — Ei _ 1 is greater than (E ,_ 1 — E,_ 3 ) 2 / 2 . 
Let us suppose that the law to be established holds good for a certain value of i. 
For the sake of brevity, I denote E;, E,_ l5 E;_ 2 , E;_ 3 , ... by N, P, Q, Pt, . . . 
We have then 
, Q(Q-l) 
R (R - 1) (R - 2) 
+ 
S (S - 1) (S - 2) (S - 3) 
1 - 2 
2.3 
2.3.4 
, P(P —l) 
Q (Q - i) (Q - 2 ) 
+ 
R (R - 1) (R - 2) (R - 3) 
2 
2.3 
2.3.4 
S (S - 1 ) (S - 2) (S - 3) (S - 4) 
2.3.4.5 
* Professor Cayley has brought under my notice a not altogether dissimilar, but perhaps less striking, 
phenomenon, pointed out by Cauchy, that, although the series 
is convergent, its square 
i.e., 
v / 2^~v / 3 P 4 ’ 
Uq + (2‘ u 0 u l) d~ (2 UqU% + Mp) + . 
* J 
1 ~ V ' 2 + (^3 + *)“ 
is divergent. 
f The fortunate circumstance of the two ratios in question being always respectively less and greater 
than the common asymptotic value of each of them enables us to find the value of the constant in the 
expression e 2 *, which is asymptotically equivalent to the half of the x ib Hamiltonian or Hypothenusal 
Number by a method exactly analogous to that of exhaustions for finding the Archimedian constant 
correct to any required number of decimal places. See end of this section (pp. 298, 299). 
2 P 2 
