ON HAMILTON’S NUMBERS. 
303 
This equation is easily transformed into 
(1 — x) E ° + x (1 — x) El + a; 3 (1 — x) E ~ + . . . + x n (l — x) En 
— 1 — 2a: -f- a: 2 (1 — x) E, ‘“ 2 F„ ( x ) — x H +1 (l — x) E " -1 , 
from which, as Professor Sylvester has pointed out in this memoir, by equating 
coefficients of all powers of x from 0 to n, we can obtain the successive values of E„. 
The general formula 
T? I -^"-2 (F «_2 — 1) , / Po(Po — !)••• (E 0 — n + 1) 
1 - K-! + J75 ■••+(-) 1.2 ... 'll - 0 
arises from equating the coefficients of x ".— J. H.'“ 
§ 3.+ Sequel to the Asymptotic Theory contained in § 1. 
The relation 
p = r — IP etc - 
previously obtained supplies only the two first terms of the remarkable asymptotic 
development 
r ~P 
= I {p + ( T + P J r • • • + 2®’) + H, 
where i is any assigned integer and H is of a lower order of magnitude than the lowest 
power of q in the series which precedes it. This may be easily established as follows :— 
By the scale of relation proved in the preceding section we have 
Let, now, 
therefore 
and 
9 , O Q I ° I 
P = r- v +« + ••• 
o 
= (f — |-r 3 + terms whose maximum order is that of r~. 
P = <f — VP - i h P - IV - ¥p • • • ; 
q — f 
P 1 - f hr* - ffo* - ifry 
p ■=. (p — -|r 3 (1 — v i — hr a 3 — kr B 2 — lr y 2 
_ |/ ir 2»- _ _ |^V . . . 
q 2 — -fr J -j- | (F + hr a+1 -f- &P* + 1 -j- If 
- ¥^ a - pr 2 * 3 - fZr 8 * - 
Tv + i 
Therefore 
)+•■• 
■) + ••• 
2 « = 
h = 1, & = 1, l = 1, m = 1, ... 
2/3 = 1 + a, 2y = 1 + & 28 = 1 + y, 
t.e., 
“ = 
/3 = f, 
v ~ i- 7 - 
y — 1 6 i 
S — £3 
° — 3 2 J 
* See Note 3, p. 312. 
f Received July 28, 1887. 
