296 
PROFESSOR SYLVESTER AND MR. J. HAMMOND 
Then 
q l — q = (r 2 — § r — rf + k (r 3 — § r — /•)“, 
or, neglecting terms of lower dimensions than r 3 , 
/ o 2 3 \o I On 
{> - 3 r - & + K1 ■ 
Therefore 
Consequently 
| r 3 = — 2r 3 + k r 
a 
— f and k = -f. 
Thus, then, for the consecutive Hypothenusal Numbers A., g, 
Let 
or say 
X — /r- -j- 3 g + • • . 
X = g 2 -f | /v + d/r, 
+1 = + I V + g. 
where g x is the x th term in the series 1, 3, 18, . . . 
The successive values of p. c and their differences are given in the annexed Table. 
X 
v* 
P* 
1 
•5 
•55719096 
2 
l 
'66666666 
+ 1094/5/0 
3 
3 
•69059893 
+ -02393227 
4 
18 
•67647909 
-•01411984 
5 
438 
•64334761 
-•03313148 
6 
204348 
•61769722 
-•02565039 
7 
41881398318 
•61139243 
-•00630479 
8 
1754062953103547795958 
•61111171 
— •00028072 
The decimal figures following those given in p s , required for ulterior purposes, 
being 5795. 
An examination of the column of differences for x = 5, 6, 7, 8, shows that the 
ratios of each to the rest go on decreasing somewhat faster than their squares : this 
makes it almost certain that p s — p 0 will be between the 400th and 500th part of 
•000280, and that accordingly the value of p g will be ’6111111, &c. I believe it is 
beyond all moral doubt that the ultimate value of p is exactly yy; and, indeed, it 
was the conviction I entertained of this being its true value, when I had calculated p : , 
