298 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
L * (F - - ~ ]) — 6153473687194529702895764001115884685871706 
= 9795094444S414216137607200637520 
1.2.3 
F 6 (E fl -lHE 6 -2) (E 6 — 3) 
1 . 2 . 3 . 4 
= 1173024302352295838445 
E 5 (E s — 1) (E 5 — 2) (E 5 - 3) (E 5 - 4) 
= 5552272910184 
1 . 2 . 3 . 4.5 
E, (E 4 - 1) (E 4 - 2) (E 4 - 3) (E 4 - 4) (E, - 5) 
= 12271512 
1 . 2 . 3 . 4 . 5.6 
E s (E s - 1) (Es - 2) (E 8 - 3) (E 3 - 4) (E s - 5) (E, - 6) 
1.2.3.4.5.6.7 
= 792 
v . -p 24-33333333 . . . 
rj 6 -r-r) 5 = 466-54794520 . . . 
rj 7 + V( , = 204951-34925714 . . . 
^ 4- rj 7 = 41881671184-54776412 . . . 
rjcj + rj 8 = 1754062953159389842293-346657805 . . . 
v / ^= 4-24264068 . . . 
= 20-92844819 . . . 
x/^. = 452-04866994 . . . 
204649-45227877 . . . 
= 41881534751-051659567667 . . . 
Finally, it is interesting to find the asymptotic value of h x and r/ x (the halves of the 
sharpened Hamiltonian and of the Hypothenusal Numbers), which are ultimately in a 
ratio of equality to each other, in terms of x. Obviously each of these is ultimately 
in a ratio of equality with M% where M is a constant to be determined. 
Let 
M — 10 s * and w* = I0 2 ' l+a . 
Then, for finite values of x, remembering that (in the preceding notation) 
p < cf and X > /x z , 
u x must be intermediate between the corresponding terms of the two series 
y] = l l, 3, 18, 438, 204348, 41881398318, 
h = 2, 3, 6, 24, 462, 204810, 41881603128, . . . . 
