ALGEBRAIC FORMULA. 287 



verfe axis, meeting the curve, and dividing the elliptic quadrant 

 into two parts, the greateft of which is that next to the vertex 

 of the conjugate axis ; then, if E denote the whole elliptic qua- 

 ■drant,the greateftelliptic arch fliall be equal to ^~~» ^"d the 



leafl = — ~^ "^'^ ; alfo the difference of the arches — i — 6- 



— difference of the femi-axes ; and the fame conclufion is rea- 

 dily drawn froni the theorem given in Art. 13. by afTuming 



<P+-^ — 1- 



17. Another obfervation I have to make upon our Formu- 

 la is, that we may hence deduce that relation between indefinite 

 arches of three certain ellipfes, which is mentioned at Article 

 10. of the preceding paper, and which was firfl obferved by 

 Mr Legendre ; but as the only ufe we have made of that pro- 

 perty was in the cafe of thofe arches being quadrants, or femi- 

 perimeters, I fhall, for the fake of brevity, treat only of that par- 

 ticular cafe. 



Supposing, therefore, the quantities e\ e\ e'", &c. to be related 

 to each other as already expreiTed in Art. 11. we have found, 

 that if 



P = (I+.')(I+0(x^-O,&c.andQ = i + ii;-^^-f-&c. 

 and E denote the quadrant of an ellipfe, of which the excentri- 

 city is e, then E = -P (i — eQJ). 



In hke manner, if E' denote the quadrant of another ellipfe 

 of which the excentricity is f', and we put 



F = (i+o Ci+o (x+o, &c. alfo 01= i + ^ + g!:;+ &,. 



for the very fame reafon E' = f P' (i— <?'Q1). 



O 2 BttT 



