Of KEPLER'S PROBLEM. 233 



order*?* or s' ; that of the fecond approximation is of the order e 6 ; 

 and in none of the planetary orbits will it be necefTary to pufh 

 the approximations further than the fecond term of the feries. 



The method of finding the arch <p, that has juft been explain- 

 ed, is very commodious in practice ; becaufe the value of 

 *? 2 fin *? cof 2 q> is eafily computed by the common tables, when a 

 known arch is fubftituted for <?. But we may, with advantage, 

 apply the method of infinite feries to the refoiution of the equa- 



rr r> 1 ^ ln m e 1 



tion. IF we put /b = 1 — e l , and z — -07-X n > we nave 



fin ? r= -^7— X (1 — s 3 + 3 zi — 12 z 6 + 55 z s — &c). 



The rule requires ftill another operation, viz. to find the 

 arch -f. For this purpofe we have the proportion 



cof — : cof — - — : : cof ® : cof I. 



2 2 Y T 



But, in the cafe we are now confidering, the arch. 4 is always 

 fmall : and, on this account, the proportion above is of little ufe 

 in practice, when any degree of accuracy is required. The rea- 

 fon is, that the common tables are not computed to a fufficient 

 number of figures for determining fmall angles from their cofines. 

 We will, therefore, prefer the other method of computing 4-, given 

 in Art. 12. which is not liable to the fame inconvenience. 



The obfervations we have now made, lead us to the follow- 

 ing rule, for computing the anomaly of the eccentric in the or- 

 bits of the planets : 



1. Compute the arch r from the formula 2r — e X fin m. 



f 



2. Compute e zz s X j and determine the arch <p from 



the equation fin *? zr fin m -f- e z fin <? cof : <?.. 



fin p n t 



'j. Compute tan A zz e X r X fee 4c ; and fin - 



J r p — in '*''-. « ■ 



cof ^ — 



2 



— tan — X fin 4c . 

 2 ^ J 



Then «, = <? — 4v 



It 



