iS8 REcriFICJTlON of the ELLIPSIS, &c. 



any term is nearly equal to the following tenii, and the rapidity 

 with which the feries decreafes is therefore very great. 



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

 tions for computing A and B, is fhortly this : 



Put c — ^ : and compute 



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



B = ^ X (i +0 X (M + N). 



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

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

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

 it will therefore not be neceffary 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 (ince found, however, that 

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

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

 ^xpreffions, 



A rr (I + c') (i + c') (l + C") &c. 

 B^.x(i + ^ + f^+f^4+&c.)XA. 



I — cof J 



1^oK if c r= fin nif then s/i — c^ zz cofm and c' = x + coim 



zz tan^ - : confequently i + c' zz fec^ ^. In like manner, if 



c' — iinm', c" — fin m\ &c. we fhall have fin m' — tan^^ ; 



finwr^ 



