i84 RECTIFICATION of the ELLIPSIS, &c. 

 Our fluxional formula thus becomes <pir 



I NEXT transform the quantity ^ i — c^ fin ^4* as in the in- 

 veftigation for the elUptic feries, and putting c' zz ^^ __^^ , I 



v/l + c'^ + 2c' cof 2\^ 



Now, taking the fluents when <p zz w, and -v^ i= w, we fhall 

 have y , , ^ n A X w ; And according to the me- 



Vl + c"- — 2<rcolp ° 



thod of M. DE LA Grange, /tt^TT^^t^^ = vr X 



(i + Ic'^ + ^^^'* + &c.) : Hence A = (i + X 

 \ ' a* '1.4 / 



/t J- i! c'^ 4- -^^^ t'* + &C.V And in this value of A, c' will 



V ' a^ ' 2". 4^ / 



be a fmall fradion, even though c be large ; and the feries will 

 therefore converge very fafl. 



But, taking the value of A diredlly in a feries, we have 



A = 1 + ~ ^^ + ^. ^* + &c. Andfo I + 11 c' -f ^c^ 



4. &c. = (i + c') X (i + i; ^'» -f ^ C* + &c.)^ Now, 



the two feries being exadlly alike, it is evident that we may 

 transform the one, as we have transformed the other, and that, if 



we put c" z=. — ■ . ,^ - we Ihall have i -j- ^ c'^ 4- ^^'^, c'* r: 

 {i +^^0 X (1 + '^c'" +^^c"^ + &c.): whence A = (1+0 



(i 4. ^//) (i + i! c'^ + i^iil c^^^ + &c.). 



It 



