224 Proceedings of Eoyal Society of Edinburgh. [sess. 
abscissas. By drawing on paper (four times the scale of the 
annexed diagram), showing engraved squares of '5 inch and 
•1 inch, and counting the smallest squares and parts of 
squares in the two areas, I have verified that they are equal 
within less than 1 per cent, of either sum, which is as close 
as can be expected from the numerical approximation shown 
in the tables and from the accuracy attained in the drawing. 
/ 
\ 
/ 
\ 
7 
V 
/ 
\ 
\ 
J 
r 
\ 
/ 
\ 
/ 
0 
5 
4 
5 
V 
6 
7 
8 
9 
■ 0 
\ 
7 
\ 
\ 
V 
7 
\ 
7 
\ 
\ 
\ 
T~ 
/ 
£ 
Fig. 2. 
§ 7. In Table I. (argument /) all the quantities are shown 
for chosen values of /, and in Table II. for chosen values 
of r. The calculations for Table I. are purely algebraic, 
involving merely cube roots beyond elementary arithmetic. 
To calculate in terms of given values of r the results shown 
in Table II. involves the solution of a cubic equation. They 
have been actually found by aid of a curve drawn from the 
numbers of col. 3, Table I., showing r in terms of r\ The 
numbers in col. 2 of Table II. showing, for chosen values of r, 
the corresponding values of /, have been taken from the curve ; 
and we may verify that they are approximately equal to the roots 
of the equation shown at the head of col. 2 of Table I., regarded 
as a cubic for r with any given values of r and K. 
