ON HAMILTON’S NUMBERS. 
299 
By means of this formula, writing for u x corresponding values of 77 and h, and 
retaining so much of the two corresponding determinations of a as is common to both, 
we can find a precisely to any desired number of places of decimals, as shown in the 
following Table, in which 18 and 24 are taken as the terms of place zero in the 
respective series. 
CO 
r-H 
11 
438, 
204348, 
41881398318, 
a = -32, 
•401, 
•4088, 
•4089863 . . 
u x = 24, 
463, 
204810, 
41881603128, 
a = -4 6, 
•413, 
•4090, 
•4089866 . . 
Hence, if we now change the origin, taking \ and 2 as the zero terms, we have 
approximately 
w x+z =\& x+a 
and 
8 log M = 2 ' 408986 , 
O 7 
which gives 
M = U4654433 . . * 
As a verification, since 2 3 = 8 , (t'46544) 8 should lie between 18 and 24; and, as a 
matter of fact, a rough calculation gives 
(U46544) 3 = 2-1473..., 
(2-1473) 3 = 4-608 ..., 
(4-608) 3 = 21-234 ..., 
which is about midway between the two limits.-- J. J. S. 
5 2 .- 
Let 
Proof of the Formula for the Successive Determination of each in turn of 
Hamilton’s Numbers from its Antecedents. 
1 + & + 
2x + 
x? 4 4 4 4 ad 4 
34 + 44 + 54 4 64 + 
6441544 2944 494 + 
364 4 2104 4 804.4 + 
4 4 • • • = F 0 (af), 
74 + . . . = F x (x), 
764 + ... = F z (x), 
24494 4 . . . = F 3 (x), 
where the coefficients of the various powers of x are the numbers set out in the 
triangular Table at the commencement of this paper. 
If, in general, we write 
F n {x) = a n x n 4 h M x n + 1 4 c u a?" + 3 4 4 at , + 3 4 • ■ • , 
the coefficients of F„ + 1 (x), expressed in terms of those of F„ ( x ), are as follows :— 
* See Note 1, p. 312. 
2 Q 2 
