ALGEBRAIC FORMULA. 263 



which is known to fubfifl between indefinite arches, as well as 

 between the whole perimeters of any three contiguous terms of 

 a feries formed by an infinite number of ellipfes, the axes, or ex- 

 centricities of which have among themfelves a rery remarkable 

 conneiflion. 



Let E, E', E", E'", &c. denote the feini-perimeters of a feries 

 of ellipfes ; e, e', /', e'", &c, their excentricities, and c, c, c", c'", 

 &c. their conjugate axes ; let thefe ellipfes be fo related to each 



other, that e' = i^ = i:^-^!^!', .'^ = L=£' = '"^lEZ!' 



«'" = TTT-' ~ - "7 /~"..[, , &c. Then we have the following; 



feries of equations *. 



(r + e')E ^ (2 + c) E' + C (i + c') E" = o, 

 (l + ^0 E' — (2 + f*) E" + c' (i +0 E'" = o»- 

 ( I + e'") E" — (2 + c"') E'" + c'" ( I ;f c'") E'^ = o, 

 &c. &c. &c- 



This feries of equations may be continued backwards by 

 putting E', E', &c. for the femi-perimeters of ellipfes of which 

 tlie excentricities are e\ e^\ &c. and femi-conjugate axes c\ c'\ 



and fince e = 1^^^^ ^v _ l^zj^^^^ ^ therefore, er' = 



II. From the foregoing feries of equations, it appears, that 

 any ellipfe of the feries E\ E\ E, E', &c. may be exprefled by 

 means of any other two ellipfes of the fame feries : for the num- 

 ber of equations is always two lefs than the number of ellipfes.; 

 and therefore, having aflumed any nvunber of equations, we 



L 1 2 can» 



• See a Memoir upon the Comparifon of EUiptic Arcs, by Legendre, in the. 

 Memoirs of the Royal Academy of Sciences for 1786. See alfo the Appendix .^ 

 to this Paper. 



