SECT. 3.] PLACES ix ORBIT. 121 



taken again from table III., will not differ from the first value ; otherwise it would 

 be necessary to repeat the calculation anew until it underwent no further change. 

 When the quantity x shall be found, g will be got by the formula sin 2 1 g = x. 



These precepts refer to the first case, in which cos/ is positive ; in the other 

 case, where it is negative, we put 



and 



ri 1^-1 MM 



whence equation 12* properly reduced passes into this, 



PI CjfeT TT \ * ~T&quot; A J * * 



[15*] ff = LJ-_2_. 



I 7 &quot; and -T can be determined, accordingly, by this cubic equation, whence again x 

 will be derived from the equation 



r~i /**n T &quot;* -&quot;* 



[16*] x = L YY- 



In the first approximation 



MM 



will be taken for H; will be taken from table HI. with the value of x derived 

 from H by means of the equations 15*, 16*; hence, by formula 14*, will be had 

 the corrected value of H, with which the calculation will be repeated in the same 

 manner. Finally, the angle g will be determined from x in the same way as in 

 the first case. 



92. 



Although the equations 15, 15*, can have three real roots in certain cases, it 

 will, notwithstanding, never be doubtful which should be selected in our problem. 

 Since h is evidently a positive quantity, it is readily inferred from the theory 

 of equations, that equation 15 has one positive root with two imaginary or two 

 negative. Now since 



m 



16 



