vox.. tX.] PHILOSOPHICAL TRANSACTIONS. TJ 



inches in a year. This would be the stroke if the emission were at its max- 

 imum. Is it not owing to the extreme minuteness of the fibres of the nerves, 

 that a stroke, which is certainly less than the toVt part of this, is not sustained 

 by our organs, without pain? 



XXXVI. Some New Theorems for Computing the Areas oj certain Curve Lines. 

 By Mr. John Landen, F.R.S. p. 441. 



Tlie learned editor of Mr. Cotes's Harmonia Mensurarum first gave us, in 

 that book, the celebrated theorems for computing the areas of the curves whose 



ordinates are expressed by — - — ,, . , , . r . . . „. , or „ , „ „_„ , — -,. ; and se- 



veral other writers have since done the like. Which theorems consist of many 

 terms, being obtained by previously resolving the expression for the ordinate, 

 into others of a more simple form. Now I have found, says Mr. L., that the 

 whole area of every such curve, when finite, may be assigned by theorems re- 

 markably concise, without the trouble of resolving the expression for the ordi- 

 nate as aforesaid ; and as in the resolution of problems, the whole area of a curve 

 is more commonly wanted than a part of it; and as these new theorems enable 

 us to compute such whole areas as above mentioned, or the whole fluents of 



-— — -, 7- — -V — T , ■ -.. > and—————— -, with admirable facility; I do my- 



self the honour of communicating them to the Royal Society, presuming they 

 may be thought worthy to be published in the Phil. Trans. ,, 



Theorem 1. m being any positive integer or fraction, and n any such integer 

 or fraction, greater than m; the whole area of the curve, whose abscissa is x, and 

 ordinate -- ;— -, is = -7- X a. ^A JmiK 



a" ■\- x"' jn 



Theorem 2. m and n being as before mentioned, the whole area of the curve, 



whose abscissa is x, and ordinate — - — r,-= .^ , _„^ is = + °~ ~ -"::- v £. 



(o"+ X") X (e" + X") — a«_e» ^ fn 



Note, when e is = a, the expression for the area becomes = ^^^-=^ — v A. 



Theorem 3. m and n being as in the preceding theorems, the whole area of 



the curve, whose abscissa is x, and ordinate -- — -p- — is = ^^ — v a. 



' a" + 2ca"x" -f x" bfn 



Note. If m be = O, the area will be = —7 — . 



on 



In these theorems, < 



A denotes the semi-periphery of the circle, whose radius is 1 ; 

 B an arc of the same circle, whose cosine is c and sine b , 



/"the sine of the arc - X A ; ^f the sine of the arc - X B. 



Concerning the investigation of these theorems, it is sufficient to say, they are 



