ON HAMILTON’S NUMBERS. 
309 
Since 
q = r 2 = s 4 (ultimately), 
the terms of Order 1 (which are the only ones with which we have to do at present) 
are ultimately equal to 
(- 26 x + 2 - f + \)q ; 
or, giving 6 X its ultimate value -g-, to or to the same order of approximation to 
U(q ~ r). 
Hence, ultimately, 
(p -?)=(?- r f + s(? - r f + xi( r i - r)* 
We use this result to obtain a closer approximation to \/ q than r — 6 X \/ r, and to 
find the relation between the general values of 6 X and 6 ' 2 . 
Thus, assuming 
v 7 q — r = r — 5 -f f v 7 r — s + k, 
we have, ultimately, 
q — r — (r — sf + 1(7* — sf + (I + 2 k)(r — s) 
= (r — sf + |(r — sf + H(r — s). 
Consequently, as r becomes indefinitely great, Jc converges to the value 
1/ 11 4 \ — 1 
2\1 8 9 / — 12 - 
Now 
V 7 q — r = %/ q — i . . . = s/ q — ^ ultimately ; 
V 7 r — s = v r — \ ultimately. 
and similarly 
Hence, ultimately, 
v 7 q — v — s + f- v 7 r -j- yy + \ — y = r — s f V 7 T + i- 
We may therefore write 
\/ q = r — s + f\/ r + e (where ultimately e = y). 
v 7 q — r— 6 X V 7 r, 
6 X v 7 r = s — f \/ ■ 
v / r = s — d 2 v 7 s, 
But 
and therefore 
Moreover 
whence it follows that 
r —- e. 
v 7 r = s + f \/ s — e (where e = \ ultimately). 
* As previously obtained by observation in § 1 (pp. 296, 297). It will, of course, be understood that in 
the above and similar passages the sign = is to be interpreted to mean “ is in a ratio of equality with.” 
