i84 RECriFlCATlON of the ELLIPSIS, &c. 

 Our fluxional formula thus becomes (p = 



^ 





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



I — / I - r* 



vefligation for the elUptic feries, and putting c' — _ ^ , I 



find /i-r^fin-«^= ^'+^';ty''°'''^ , and fo . ^ / = 



^I + c' ' + 2c' cof 24. 



Now, taking the fluents when <p =. w, and ■^ ■=. ■a^yre, fhall 

 have y^ , , '' 3: A X » : And according to the me- 



■v/l+c' 2CC0lp *> 



thod of M. DE LA Grange, /;^rT7T=^^^^ = «r X 



(i + llf'^' + ^v^c'^ + &c.) : Hence A = (i -f <:') X 



fi^llc"' + ^^T^ C* + &€.)• And in this value of A, c' will 



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

 therefore converge very faft. 



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



A = I + ;4 ^= + ^ f * + &c. Andfo I + 11 ^^ + ^c* 



+ &c. = (i + f') X (i + ^ c'^ + i;ii; ^'4 + &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" ~ '~^'~^,. we fliall have i + il f'= + ^^ c'* - 

 (i + c") X (1 + ;-; c"^ +^f"'* + &c.):whenceA = (i +<) 



{i + f") (i + ^ '^"' + ^ c"^ + &c.). 



It 



