v88 REcriFICjTlON' of the ELLIPSIS, &c. 



anytei'm is nearly equal to the following temi, and the rapidity 

 with which the feries decreafes is therefore ver}' great. 



The method, then, that refults from the preceding inveftiga- 

 tioiis for computing A and B, is fhortly this : 



Put c' — ' : and compute 



and V + fi-3 ,.3 + 1^.1^ ,. + &,. ^ N. 



Then A = (i + c') X M, and 



B = c X (i +c')x (M + N). 



The feries M and N will converge fo faft, even in the moft 

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

 the firft three terms will give the fums fufficiently exadl ; and 

 it will therefore not be neceflary to have recourfe to the more 

 converging feries A" and B". 



Such is the method that I had firft imagined, for facilitating 

 thefe fort of computations. I have fince found, however, that 

 by means of the common tables of fmes and tangents, the quan- 

 tities A and B may be computed in a ftill eafier way from the 

 €Xpreflions, 



A = (I + c') (i + c') (i + O &c. 



B = .x(r + ^'+{4'+|4'$+&c.)xA. 



For if c = fin m, then v/i — c* zz. cof «z and c' =. ^ ^ ^^^- 



zz tan ' - : confequently i -\- c' zz {tc^ —. In like manner, if 



r' zz fin ra', c" zz fin 7/1", &c. we fliall have fin m' zz tan ' ^ ; 



fin?«* 



