¡4 THE INVESTIGATION BY WALLIS OF HIS EXPRESSION FOR 77. 
C. XIII. 
the sixth line is 
r. 6 
A, 1, Aj, l.|, 
5/7 6^8 
1.3’ 2.4’ 
. 5 A .9 6.8.10 
^1.3.5’ '2.4.6’'"’ 
and this being so, he completes the table by inserting a term □ in the diagonal 
line between the 1 and 2, and calculating the odd lines as follows: viz. if in each 
case A is the first term and B the second term of the line, then the complete rule is: 
the first line is 
the second line is 
A, 1, 
the third line is 
A, 1, 
the fourth line is 
4 
l. 
l 
'2’ 
, 0.2 
A 1.3’ 
1. 
1.3 
2.4’ 
A 
4- 
l 
2 
'2’ 
CO CO 
r—1 l>—1 
1 
2.4 
‘ 2.4 ’ 
A 
4. 
l 
3 
'2’ 
A 2 ' 4 
A 1.3’ 
1 
3.5 
‘2.4’ 
A 
0.2.4 
1. 
1.3.5 
A 
A, 1 
the fifth line is 
A, 1, A 
3.5 
1 _ A ZlA 
1 '2 ’ A 1.3 ’ 
1. 
4.6 
1. 
2.4’ 
A 
1.3.5’ 
1. 
1. 
1. 
1. 
2.476 ’ 
and so on, the rule for the odd lines being thus an interpolation from that for the 
even lines. Observe that, in the first line A is = oo, and that, in the several odd terms 
after the first, A ^ is =1. 
The table thus calculated (I have given here the fractions in their least terms) is 
oo 
1 
1 
2 
3 
8 
T 4 5° 
5 
T w 
□ 
•fe 
3 5 
12 8 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
□ 
3 
2 
!□ 
1 5 
~s 
!□ 
35 
T6" 
n* 
315 
12 8 
1 
3 
2 
2 
1 
3 
4 
9 
2 
5 
1 
1° 
~2 
1° 
3 5 
“8" 
f|D 
10 5 
"T(T 
12J. n 
2 1 u 
115 5 
128 
3 
8 
1 
1 5 
8 
3 
35 
8 
6 
6 3 
8 
10 
9 9 
8" 
15 
1 
7 
2 
ff D 
63 
8 
-W-D 
2 31 
T6 
3 0 0 3 
12 8“ 
T(T 
1 
3 5 
T17 
4 
10 5 
10 
231 
16" 
20 
4 2 9 
"T6" 
35 
1 
3 5 U 
9 
2 
I_28,n 
2 1 U 
9 9 
8 
W D 
4 2 9 
16 
4rf 4 ° 
6435 
12 8 
35 
12 8 
1 
315 
12 8 
5 
1155 
12 8 
15 
3 0 0 3 
12 8 
35 
6 43 5 
12 8 
70 
in which ta 
4 
the value — 
7T 
The thir 
The suc< 
manner the 
to have beei 
- > -, that is 
* V 
and these equ; 
limits which te 
the number of 
denominator, ar 
factors. 
It is to be 
£, 1, f, ... as 
thus m = n = \ 
n (|) -7- n (3) n