SECT. 3.] PLACES m ORBIT. 115 



Let us put 



2 cos/ 



and then will 



a 



k. * 



also 



./ a = _|_ 



sin 5- 



in which the upper or lower sign must be taken, as sin y is positive or negative. 

 Formula XII., article 8, furnishes us the equation 



- =E e&a.E E + e sin.E = Zg 2 e siny cos G 



a 3 



= 2ff sin 2 y -j- 2 cos f sing ^ . 



If now we substitute in this equation instead of a its value from 10, and put, for 

 the sake of brevity, 



we have, after the proper reductions, 

 [12] m = 



in which the upper or lower sign is to be prefixed to m, as sing is positive or 

 negative. 



When the heliocentric motion is between 180 and 360, or, more generally, 

 when cos/ is negative, the quantity m determined by formula 11 becomes im 

 aginary, and I negative ; in order to avoid which we will adopt in this case, instead 

 of the equations 9, 11, the following: 



_ 



2 cos/ 



- - *i-^ - -=M, 

 2*( cos/f (rr f 



whence for 10, 12, we shall obtain these, 



