RECTIFICATION of the ELLIPSIS, &c. 183 



In confidering thefe fubje<f);s, 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 required with a fufficient degree 

 of exa(5lnefs. This is what I am now to communicate. 



We are then to confider the expreffion (<2^4-^^ — 2^cof(p) 

 n , ^ : for the fake of fimplicity in calculation, 



I write - zz Cj throwing out a altogether ; and I fuppofe 

 I ^ r: A + B cof (p + C cof 2(p + &c. 



Let %// be an angle, fo related to (p, that fin (^^ — (p) rr c fin %|/ ; 

 It is obvious, from this formula, that -^h = <p when fin ■^ = o^ 

 that is, when -^ is equal to o, or to w, 2vr, &c. 



We have then, cof (^j/ — <p) =Vi — c^ {m'^P: and taking 



. __ ccof-^y.4' c cof v}/ X 4- 



the fluxions, ^^ — cp — cofc,).— 4.) — ■/(! — <^^ fin ^4-) * 



^ I — c' fin ^4- — ccof 4/ 



whence (pr= 4/ X ' y'x — c^ im ^4- 



But (/i — <7' hn'-vj/ — c cof ^^)^ =: i — c' fm'^|/ -f c^ cof "-^z 



2C Cof4 V' I — C"" fin ^-v// ZZ I + r^ 2C' fin ^-v^ — 2C cof%i' 



y' I __ ^ 2 fin 2v|>, (becaufe r^cof ^-v// zz^* — c* fin^-v^) rr i + ^^ 

 — 2cX (c{m''^ X fin%// + cofi// v'l — c^ fin =4-). Now, if we 

 write for <7fin 4. its equal, fin (4/ — (p), and for 1/1 — <7^ fin ^xj/ 

 its equal, cof (oj/ — <?), we fhall hav« c {m^P X fin ^^ + cof ^^ X 

 Vi _ c^ fin '4 = fin(4z — <p),X fin "4/ + cof^/ X cof(4/ — <p) 

 iz: cof (p : which being fubflituted, there comes out 

 (/i — c' fin*^' — c cof 4')' = !+<:* — 2C- cof (p. 



GUR. 



