[ 44-2 3 
admirable facility ; I do myfelf the honour of com- 
municating them to the Royal Society, prefuming 
they may be thought worthy to be publilhed in the 
Philofophical Tranfadtions. 
THEOREM I. 
m being any pofitive integer or fradtion, and n any 
fuch integer or fradtion, greater than ;;; ; the whole 
area of the curve, whofe abfciifa is x , and ordinate 
X">— I . a m — n , 
IS = —r— X A. 
a n -\-x n 
f* 
THEOREM IL 
m and n being as before- mentioned, the whole 
area of the curve, whofe abfciifa is x 3 and ordinate 
I . a + m e -\-m A 
-- -■ ■: — ig -f- — ^ V 
a r - -f- x n X t n + xn — a n — e n fn 
Note. When e is —a, the exprefiion for the area 
, rna J r m — n . 
becomes = x . x A. 
/«* 
THEOREM III. 
m and n being as in the preceding theorems, the 
whole area of the curve, whofe abfciifa is x y and 
y n-\-m — i _ g q-\' m — n 
ordinate 
u' ln -)r ica n x n x- n 
bfn 
X A. 
Note. If m be=0, the area will ber= 
In thefe theorems, 
a — "B 
b n 
A de- 
