QUADRATURE of the CONIC SECTIONS, &c. 297 



By fubftituting thefe transformed exprefEons in the feries, it 

 becomes 



3 13 C0fA A + 12 Cof2 A , 14 



A + 3+coi4A — 4 col 2 A 15 



1 



{1 13 — cof2 A — 12 cof A , _i_ 13—- cof A — 12 cof 7 A 

 16 3 + cof 2 A + 4 cof A + ¥ ! 3 + cof A -j- 4 cofi A 



4. * 13 — cofjA — i2cofjA , & "I 

 "•"I6 3 3 + cofiA + 4cof^A ^' J 



-Finally, let ~ a be now fubftituted for A, and -§■ a for ~ A, 

 and fo on, and let the refult be divided by 3X16; then we 

 have 



I 1 13 — cof a +12 cof j a , 7 



a 4 ~ 3.16 2 2 + cof a — 4cof-r« 8.8.0.10 



-I 





ri 13 — cof 7 a — 12 cof * 7a , 1 13 — coC^a — i2cofja 



3.163 3 -J- cof \a + 4 cof % a 3.16* 3 + cof^^H- 4cof£> 



4. * T 3~ cofja — 12 cof -V^ . &c ^-i 

 3.16 5 3 -j- cof j a 4- 4 cof T 'y # - 1 ' 



which is our fourth general feries for the rectification of an 

 arch j and its rate of convergence is very confiderable, for each 

 term is lefs than, ^th of the term before it. The feries, how- 

 ever, approaches continually to a geometrical progrellion, of 

 which the common ratio is ^. 



29. The preceding formulae, as well as innumerable others, 

 which, may in like manner be deduced from the expreflion 



tan A =z — - — - tan \ A, all agree in expreffing a power 



of the reciprocal of an arch by an infinite feries, the terms of 



which 



