[ 44i ] 
Received May 14, 1770. 
XXXVI. Some New 'Theorems for computing 
the Areas of certain Curve Lines: By 
Mr, John Landen, F.R,S. 
Read June 28, y A '1 H E learned editor of Mr. Cotes’s 
Harmonia Menfurarum firft gave 
us, in that book, the celebrated theorems for com- 
puting the areas of the curves whofe ordinates are 
exprelled by — — , - ■ -^ ,or — — : ; 
* J + X n xe n + x n a 2n -{-2ca n x n -\-x^^ i 
and feveral other writers have fince done the like. 
Which theorems confifl: of many terms, being 
obtained by previoufly refolving the expreffion for 
the ordinate, into others of a more Ample form. 
Now I have found, that the whole area of every 
fuch curve (when finite) may be afiigned by theorems 
remarkably concife, without the trouble of refolving 
the expreffion for the ordinate as aforefaid : and, as 
in the refolution of problems, the whole area of a 
curve is more commonly wanted than a part thereof ; 
and as thefe new theorems enable us to compute fuch 
whole areas as above-mentioned, or the whole fluents 
r xP X xp x , Xpx . , 
^ a n + x ni a n + x n xe n + xn 3 a ln + 2ca n x n +x*” i W1 
Vol. LX. L 1 1 admirable 
