RECTtlFICJTlON of the ELLIPSIS, &c. 183 



In confidering thefe fubjecfis, however, I have fallen upon a 

 method of computing the quantities A and B for the exponent 

 — ^ by feries that converge fo fafl, that, even taking the moft 

 unfavourable cafe that occurs in the theory of the planets, two 

 or three terms give the values reqinred with a fufficient degree 

 of exaflnefs. This is what I am now to communicate. 



We are then to confider the expreffion (<3^-j-^^ — zaicoftp) 



= v/a'+^'-2«3cofy = ^""^ ""^^ ^^^^ °^ fimplicity in calculation, 



I write -■=.€, throwing out a altogether ; and I fuppofe 



^, ^ / f— = A + B cof (p + C cof 29 + &c. 



Let •v// be an angle, fo related to (p, that fin (i^ — ip) =r f fin ■v// 1 ^ 

 It is obvious, from this formula, that 4/ = <p when fin 4- — o, 

 that is, when -^ is equal to o, or to w, its, &c. 



We have then, cof (-v^ — <P) =-Vi — c" fin = i|/: and taking 



- . ■ • __ f Cof rJ-XJ- C Cof >;> X -j, 



the fluxions, 4. — <f) — cof(4,_,},) — y'Ci — ^^ fin-+) • 



• , . j/i — c' (in 'ij- — rcofj/ 



whence <p = 4/ X ^ ^i_^. im =^ 



But [Vi — C fin =4/ — c cof %|/)^ = i — c' fin'i]/ + r' cof '^z 

 — 2Ccof4 v'l — <:Vfin ^4/ = I + f ' — 2c' fin=-i|/ — 2c cofi|/ 



Vi — f^ fin2<J', (becaufe f'cof^i// =t' — r^fin^^-) = i + c"" 

 — 2cX (cfin^fz X fin 4/ + cof^/ V \ — r' fin '4'). Now, if we 

 write for rfin^/ its equal, fin (if' — <p), and for Vi — c^ fin ^(j' 

 its equal, co(f (i|/ — (p), we fhall have c fin -i|/ X fin a// -f cof 4' X 

 v^i — C" fin =4' = fm (^|' — (p) X fin^/ + cof 4/ X cof(4/ — ip) 

 =: cof (p : which being fubllituted, there comes out 

 (/i — c= fin'v^ _ c cof 4-)= = I + c^ — 2C cof<p. 



Our 



