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 very remarkable 

 connection. 



Let E, E', E /7 , E'", &c. denote the femi-perimeters of a feries 

 of ellipfes ; <?, J, e\ <?'", &c their excentricities, and c, c, c\ c" r f 

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



other, that e =T ^ = TT y r=7 ,e _ — - x + ^ _^., 



e'" — l ~ c . ~ x ~~ . ~~ ^=r; &c. Then we have the following 



feries of equations *. 



(1 + f) E — (2 -f E' + C (1 + <•') E* = o, 

 ( x _}_ ^) £' — (2 + £") E' + t' ( 1 + c") E'" = o,. 

 ( 1 + e") E" — (2 + c" 1 ) E'" + 6'" ( 1 -jr c'") E IV = o, 

 &c. &c. &c- 



This feries of equations may be continued backwards by 

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

 the excentricities are e\ e"\ &c. and femi- conjugate axes c\ S\ 



and fince e = £=S^ > t = ;~g5g , See; therefore,,- = 



S£>=E!g & c . 



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

 any ellipfe of the feries E v \ 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 affumed any number of equations, we 



L 1 2. can^ 



* See a Memoir upon the Comparifon of Elliptic Arcs, by Legendre, in the. 

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

 to this Paper, 



