894] THE INVESTIGATION BY WALLIS OF HIS EXPRESSION FOR 77. 25 
C. XIII. 
4 
in which table □, as the term interpolated between the diagonal terms 1, 2, denotes 
4 
the value — as before. 
77 
The third line of the table is 
1 n 3 4 3.5 4.6 3.5.7 4.6.8 n 3.5.7.9 
2 U 1, U, 2 , gU, 2 4 , g 5 u > 2.4.6’ 3.5.7 U ’ 2.4.6.8 ’ 
The successive even terms continually increase but tend to equality, and in like 
manner the successive odd terms continually increase but tend to equality; it seems 
to have been assumed that, for any three consecutive terms x, y, z, we have 
- > ^, that is, y 2 > xz. Taking this to be so, we have 
l>-rD 2 , D 2 > 
32 >ia 2 , |n 2 > 32 - 5 
2’ 2 2 3 
3 s 
3 2 . 5 2 4 2 .6 r-, 
> ^^7 0 2 , 
22.4’ 2 2 .4 2 3 : 
and these equations give □ 
less than 
greater than 
3 2 
2.4 \/ 
/4 
3’ 
3 2 .5 2 /6 
2.4 2 .6 V 5’ 
3 2 .5 2 .7 2 /8 
2.4 2 .6 2 .8 V 7’ 
3 2 .5 2 /7 
2.4 2 .6 V 6’ 
3 2 .5 2 . 7 2 /9 
2.4 2 .6 2 .8 V 8 
limits which tend continually to equality. We thus have 
4 _3.3.5.5.7.7... 
7r 1 2.4.4.6.6.8...’ 
the number of factors in the numerator being always equal to the number in the 
denominator, and the accuracy of the approximation increasing with the number of 
factors. 
It is to be remarked that for a square; row m and column n; m or n = — 0, 
^, 1, f, ... as before; the term of the square is in general II (m + n) -=r II (ra) TI (n); 
4 _ 
thus m = n—\, the term is II (1) {II (£)} 2 , = 1 -f- (^7r) 2 = - , = □ ; m — 3, n = \ it is 
II (f) -r- II (3) II (^-), = \ • f • f + 6, = ; and so in any other case.