15° 



method pointed out by Mr. Landen's difcovery that an hyperbolic arc may 

 be reftified by means of two elliptic arcs, he derived the above-mentioned 

 conclufion. This theorem enables us to derive the circumference of a very 

 cxcentric ellipfe from the circumferences of two ellipfes, the excentricities of 

 which may be as fmall as we pleafe. 



Let b, b,' b" &c. be the femi-axes minors of a feries of ellipfes, the femi- 



axes majors of which are unity ; fo that b'= 4^ , b = 4^' , &c. Then 



the rectification of two adjacent ellipfes of this feries being known, the reft 

 are eafily had by the above-mentioned theorem. Let E, E', E" reprefent 

 quadrants of three adjacent ellipfes of this feries, the refpective excentricities 

 of which are ^ / ^ then 2E' -{i-^e') E = i -f- e . 4 E" — 2 ( i +e ) E' 



I — e^ 2 I — e" 



The terms of the feries b, b', b" rapidly approach to unity, fo that the 



rectification of a very escentric ellipfe, is reduced to the reftification of two 



of fmall excentricity to be performed by the common theorem. 



But when the ellipfe is very excentric, a feries may be obtained of as eafy 

 application as the common feries, and therefore is to be preferred to the 

 above methods. 



This feries is given by Legendre, and is derived by him from an appli- 

 cation of the remarkable and elegant theorem difcovered by Count Fag- 

 nani. The methods Legendre ufed to obtain this feries and its law are 

 ftrikingly ingenious, and probably will not admit of improvement. For the 

 method and law of the feries I refer to the memoir.* 



The 



* Mem. Acad. 1786. 



I learn from a very ingenious memoir of Mr. Wallace (Edinb. Tranf. vol. 5 p. 267} in 

 which the general reftification of the ellipfe is particularly treated of, that Mr. Euler gave the 

 fame feries in a work which 1 have not feen, entitled, " Animadverfiones in reftificationem 

 " eUipGs." Mr. Wallace alfo, in his memoir, has given an elegant formula for the reflification 

 of an ellipfe. 



