ON HAMILTON’S NUMBERS. 
311 
Here the terms of Order f are ultimately equal to 
+ U+¥0i)?*, 
which, when 6 1 and e receive their ultimate values, ^ and becomes 
/_4_1 _l i I 2.0 \ n i _ 10 a i # 
\ 8 1 1 T 3 T 27U — 8 1 '1 • 
From this it follows immediately that (rejecting terms of an order of magnitude 
inferior to that ( q — r) 1 ) 
P-q=(q-rf + i(q- r) ? + H(g- r) + ^ (q — r)\ 
The law of the indices in the complete development is easily deduced from the 
relation 
P 
i , q (%q — 1) r (2 r - 1) (2 r - 2) , s (2s - 1) (2s - 2) (2s - 3) 
— 2 ~T~ 9 • > “r 
2.3.4 
The terms carrying the arguments 
^2 ^ /y% ^4 ^3 ^2 ^ ^5 
furnish the indices 
9 1 3 -| 1 i _3 i JL _5 
•"> -*■} 23 j 2 ) •*-? 4 ? 2 > 4 ’ 8 > • • •» 
which, arranged in order of magnitude, become 
9 3 . 1 3 5 1 3 5 1 
"J 2 3 X 3 4’ 83 2> 83 163 4» • • • 
Thus, calling p ~ q and q — r y and x respectively, the expansion for y in terms 
of x will be of the form 
2m+1 
y = t,Kx 2 ” , 
where n has all values from 0 to oo , and 2 to -j- 1 does not exceed « f 2, he., to has 
all positive values from 0 to n] 2 or (n + 1), according as w.is even or odd. 
But, besides this expressed portion of the development of a Hypothenusal Number, 
say 7)x+i, as a function of its antecedent, r/ x , there will be another portion, consisting of 
terms with zero and negative indices of y x having functions of x for their coefficients, 
which observation is incompetent to reveal, and with the nature of which we are at 
present unacquainted. The study of Hamilton’s Numbers, far from being exhausted, 
has, in leaving our hands, little more than reached its first stage, and it is believed 
will furnish a plentiful aftermath to those who may feel hereafter inclined to pursue 
to the end the thorny path we have here contented ourselves with indicating, which 
lies so remote from the beaten track of research, and offers an example and suggestion 
of infinite series (as far as we are aware) wholly unlike any which have previously 
engaged the attention of mathematicians. 
J. J. S. and J. H. 
* Agreeing closely with, what had been previously found by observation in § 1 (p. 297). 
