RECTIFICATION of the ELLIPSIS, &c. 179 

 we have « = «, /3=:''-"~\ 7 = "-"—^ " — ^ &c 



1.2 1.2.3 



Then multiplying thefe two feries together, and putting 



r r ■ ■ . +fnp 1/ — I — mfo^ — i 



2colOT(p for Its inaaginary value c +c 'we 



Ihall find, on equating the terms, 



and fo on. 



Of the feveral feries for A, B, C, &c. the firfl deferves parti- 

 cular attention, on account of the fimplicity of the law of its 

 terms. It deferves the more attention, too, that the whole fluent 



j (p {a^-{-b'' — 2ab cof(p)", generated while (p from o becomes 



=: w, half the circumference of the circle, is =: A + ar : all the 

 other terms of the fluent then vanifliing. 



Suppose now, in an ellipfis, the femi-tranfverfe =: i, the ex- 

 centricity =: e, and (p an arch of the circumfcribing circle, rec- 

 koned from the extremity of the tranfverfe : then the fluxion 

 of the correfpondent arch of the ellipfis, cut off by the fame 

 ordinate, will be = (p / i — e' co£^(p. 



In this expreflion, I write -+- co€ 2(p, for cof.'^: and put 



the refult, (p • 1 — T — T cof 2<p = (p /«» -j- i^ — 2ad cof 2(p, 

 a and b being indeterminate quantities. 



To determine a and b, we have «' -{- 3' = i — — and 2ab — —• 



whence a -\- b=z i^und a — b =z V i — %' fo that« = ^ + 1/' — ^^ 



2 



and b - ^-V^-^\ 



Y2 T 



