44 



General Method for deiermining 



2di We shall determine by the astronomical tables the 

 longitude of the earth seen from the sun at the instant 

 which we have chosen for our epoch : let A be the longi- 

 tude, R the corresponding distance from the earth to the 

 sun, and R' the distance which answers to the longi- 

 tude QQP + A, of the earth : we shall form the four equa- 

 tions 



+ SiZx.cos {A—oC) + R"- ., .[\) 



y- 



'0 

 i?.sin (A — a) 





r, „ / rt'.sinfl.cos 9 1 

 y=-x.|A.tangS+^^ + ^^ 1 



i? . sin 9 cos S 



..(3) 



= y^ 4- a^.x'- 



+ {y-tans9+ J^J 



l).cos {A—oi.) — 



cos^ 

 sin(y/ — a)' 



R 



.(1) 



^ c,ax.[{R' -\).sm{A-a.) + ^^\ 



\_ _2_ 

 '^"K* ~ r ' 



In order to draw from these equations the values of the 

 three unknown quantities x^ y, and r, we shall begin by 

 considering if, abstraction being made of the sign, I is 

 greater or less than I; in the former case we shall make 

 use of the equations (l), (2) and (4) ; we shall form a 

 first hypothesis for x, by supposing it, for instance, equal 

 to unity ; and we shall extract from it by means of the 

 equations (1) and (2), the values of r and y ; we shall af- 

 terwards substitute these values in the equation (4), and if 

 the remains are null, it will be a proof that the value of x 

 has been well chosen ; but if the remains are negative, we 

 shall increase the value of a;, and diminish it if the remains 

 are positive. We shall also have by means of a small num- 

 ber of trials, the true values of x, y, and r- but as these 

 unknown quantities may be susceptible of several values, 

 we must choose that which satisfies precisely or nearly to 

 the equation (3). 



In the second casr, i. e. if we have I-7 h, we shall make 

 use of the equations (l), (3) and (4), and then it will be 

 the equation (2) which v/ill serve as the verification. 



Having 



