n6 Mr. Hellins on the Rectification 
an algebraic quantity, and the arch of another ellipsis which 
is more eccentric. And as this has been done by some late 
writers on the rectification of the ellipsis,* I shall, on the pre- 
sent occasion, only state and use the result, in a notation con- 
venient for an arithmetical calculator. Thus, putting f 
, j e + uu— J iee— uu)(i— uu) 
and vv — — — — — — — — 
ze 
and the fluent of — — E, 
v(i— ««) 
- of — — F 
V(i— uu) X v'C * — 6£ uu ) ’ 
- of — N E, it is certain that 
V ( i — vv 
F is = — [l±l)u + — E— — 'E. 
2 \I — £ ] * l—n I' — £ 
Multiply this equation by i — se, and write 2s for its equal — 
(i + 0 % and it becomes 
( i— se) F 2s u 4 2E — 2 (i 40 X E. 
If we now take the fluents of the fluxions in the equation 
marked (os) in the preceding Article, we shall have 
V = E — ( i— ss ) F 4 ”• And since all these quantities begin 
and increase together, this equation needs no correction. And 
by writing for (i— ss) F its value found in the preceding 
equation, we have V = - 2 s m — E 4 2 ( 14s) 'E4 — ; and 
thence by reduction and transposition, H = 2s e u + eV+e 
E — (14 s) E'; which expression will be found to exceed 
that given (for the same purpose) by Mr. Woodhouse, in p. 
261 of the Philosophical Transactions for the year 1804, in 
the ratio of e to 1. 
* See M. Lacroix’s Traite du Calcul Differentiel et du Calcul Integral, Tome 
II. p. 1 8 1 . 
