SECT. 3.] PLACES IN ORBIT. 123 



in the seventh figure. The smaller values, however, of h are much the more fre 

 quent in practice. 



The solution of equation 15, when h exceeds the limit of the table, as also 

 the solution of 15*, can be- performed without difficulty by the indirect method, 

 or by other methods sufficiently known. But it will not be foreign to the pur 

 pose to remark, that a small value of g cannot coexist with a negative value of 

 cos/, except in an orbit considerably eccentric, as will readily appear from equa 

 tion 20 given below in article 95.-}- 



94. 



The treatment of equations 12, 12*, explained in articles 91, 92, 93, rests upon 

 the supposition that the angle g is not very large, certainly within the limit 66 25 , 

 beyond which we do not extend table III. When this supposition is not correct, 

 these equations do not require so many artifices; they can be most securely 

 and conveniently solved by trial ivithout a change of form. Securely, since the value 

 of the expression 



2 g sin 2 g 

 sin 8 $r 



in which it is evident that 2y is to be expressed in parts of the radius, can, for 

 greater values of g, be computed with perfect accuracy by means of the trigonomet 

 rical tables, which certainly cannot be done as long as g is a small angle : c&amp;lt;m- 

 venieiitli/, because heliocentric places distant from each other by so great an interval 

 will scarcely ever be used for the determination of an orbit wholly unknown, while 

 by means of equation .1 or 3 of article 88, an approximate value of g follows 

 with almost no labor, from any knowledge whatever of the orbit : lastly, from an 

 approximate value of y, a corrected value will always be derived with few trials, 

 satisfying with sufficient precision equation 12 or 12*. For the rest, when two 

 given heliocentric places embrace more than one entire revolution, it is necessary 

 to remember that just as many revolutions will have been completed by the eccen 

 tric anomaly, so that the angles^ E, v v, either both lie between and 360, 



| That equation shows, that if cosf is negative, cp must, at least, be greater than 90 g. 



