Of KEPLER'S PROBLEM. 233 



orders* or s' ; that of the fecond approximation is of the order s* ; 

 and in none of the planetary orbits will it be neceffary to pufli 

 the approxiinations further than the fecond term of the fcries. 



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

 ed, is very' commodious in pracftice ; becaufe the value of 

 £" fin (p cof ^ (p is eafily computed by the common tables, when a 

 known arch is fubftituted for 9. But we may, with advantage, 

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



TT n ^ fin 771 e 



tion. IF we put [o — i — e-, and z zz -n7~X n , we have 



fin f =: ^^— X (i — 2'- + 3 z* — 12 z" -i- 55 z"^ — &c.). 



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

 arch 4'. For this purpofe we have the proportion 



cof ^ : cof — - — : : cof ^ : cof ' . 



2 2 ^ ' 



But, in the cafe we are now confidering, the arch ^f is always 

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

 in pra(ftice, when any degree of accuracy is required. The rea- 

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

 number of figures for determining fmall angles from their cofiues.. 

 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. Compote the arch r from the formula 2t =: s X fin ?//. 



2. Compute e zz t X ; and determine the arch <t> from 



r 



the equation fin p n fin ot -f- e"- fin <P cof ?>. 



fin 9 n T 



'1. Compute tan A — ,f x X fee 45"° : and fin - 



col -. — 



2 



A . 



= tan _ X fin 4?°. 



Then u, — (? — 4'. 



It 



