ALGEBRAIC FORMULA. ' 289 



and dividing the whole equation by (1 — e ff ) } we at laft get 



( l + e') E — ( 2 + c) E' +• c\ 1 + c') E"" = o, 

 the very equation which we propofed to inveftigate. 



18. The laft remark I fhall make at prefent, upon the formu- 

 la given in this appendix for the rectification of the ellipfe, re- 

 lates to the quicknefs of its convergency ; and this, it appears, 

 will in every cafe be very great. For we have feen (Art. 7.) 

 that e' is lefs than the fquare of the excentricity, t" lefs than its 

 fourth power, e'" lefs than its eighth power, and fo on, proceed- 

 ing according to the geometrical progreflion 2, 4, 8, 16, 32, &c. ; 

 therefore, if only the three quantities e\ e ff , e'", be computed, to- 

 gether with the correfponding arches <p', <p'\ <p'", and fubftituted 

 in the formula z — 4P(i — eQ^J + &, rejecting all the factors 

 of P, and all the terms of the feries Q^and R, which involve the 

 quantity e lv , &c. the part of the formula thus rejected will be 

 multiplied by the 3 2d power of the excentricity; but if the 

 terms involving e lv be taken in, the terms rejected will be mul- 

 tiplied by the 64th power, and fo on ; thus it appears, that the 

 method here given for the rectification of the ellipfe is appli- 

 cable to every cafe of excentricity, and to every length of an 

 arch that can poflibly occur in calculation. Now thefe are ad- 

 vantages which no other individual feries, with which I am ac- 

 quainted, poffefTes ; for the common feries, which converges by 

 the powers of the excentricity, is of no ufe when the excentrici- 

 ty is great, and it is then necefTary to have recourfe to a feries 

 converging by the powers of the lefTer axis. But the feries of 

 this form, which is commonly known, converges only for a cer- 

 tain portion of the elliptic, arch, and beyond that portion it di- 

 verges ; fo that if it be required to rectify an elliptic arch be- 

 yond a certain limit, it is necefTary to find the length of the 

 whole quadrant, and thence, by the aid of Fagnani's theorem, 

 to find the length of the arch which was propofed. 



19. 1 SHALL 



