ON HAMILTON’S NUMBERS. 
297 
that led me to undertake the very considerable labour of ascertaining the 10 th 
Hamiltonian Number in order to deduce from it the value of p s . This being taken 
for granted, # we may proceed to ascertain a further term in the asymptotic value of 
7j r + j expressed as a function of q j: . 
For, calling 
fh: — x! = S* and vV = ( h> 
we have 
8 g = -00658611, 
§ 7 = -00028132, 
S 8 = -0000006047, 
9 e = 2 L "1 
q 7 = 452, > neglecting decimals. 
9s — 204649, __ 
Thus 
(Sq%= -1383, 
(Sq) 7 = *1272, 
( 8 < 7) 8 = -12375 
The value of 
and of 
(Sg ) 6 — (S }) 7 being ' 0111 , 
(Sg) 7 - (&z) 8 -0035, 
we may feel tolerably certain, from the Law of Squares, that ( Sq) 8 — (Sq) 9 will be 
somewhere in the neighbourhood of the tenth part of "0035, and accordingly that 
(S q) 9 is about "1234, so that the probable value of ( 8q ) a> is "1234 . . . 
Thus we have found 
V *+1 = y* 2 + I Vy + i i V* + [ ] V* + • • • » 
the only moral doubt being as to the degree of closeness of propinquity of the 
coefficient of r)J to the decimal "1234 . . . t 
For the benefit of those who may wish to carry on the work, I give the following 
numerical results which have been employed in the preceding arithmetical determina¬ 
tions :— 
x ' It is reduced to certainty in the supplemental 3rd section. 
t The exact value of the coefficient of rj x i, left blank in the text, is pi-oved in section 3 to be pf-, ne., the 
recurring decimal d23456790. 
2 0 
MDCCCLXXXVTI.—A. 
