RECriFICJriON of the ELLIPSIS, &c. 179 



I. 2 !• 2- 3 



Then multiplying thefe two feries together, and putting 



2 col m(p for Its imaginary value c -\- c ' we 



fhall 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 



( (p [a" + 3* — ^ah cof (p)'' , generated while <p from o becomes 



— Z3-, half the circumference of the circle, is = A + w : 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 ofl' by the fame 

 ordinate, will be =: (p / i — s' coP(p. 



In this expreflion, I write ^+~- cof2(p, for cof.'^: and put 



the refult, (p / i — T — "T ^^^^'P — ^ ^^' -{• b"- -— lab cof 2(p, 

 a and h being indeterminate quantities. 



To determine a and b^ we have -«* -f- 3* n: i and labzz^- 



a * 2 * 



whence a-\-bz=. i, and a — bz=. V i — • g' fo that a z= !. + 1/ ' -- ^^ 



2 



_ I — y/ I— £* 



- ^ • 



Y2 



and ^ =: 



2 



