154 RELATIONS BETWEEN SEVERAL [BooK I. 



likewise, according to the third solution, we find 8 by equation 



and, somewhat more conveniently, if the angles fi, /3 , are given immediately, and 

 not by the logarithms of their tangents : but, for determining i, recourse must be 

 had to one of the formulas 



Finally, the uncertainty in the determination of the angle 



X a, or iX+U 8, 



by its tangent will be decided so that tant may become positive or negative, 

 according as the motion projected on the ecliptic is direct or retrograde : this 

 uncertainty, therefore, can be removed only in the case where it may be ap 

 parent in what direction the heavenly body has moved in passing from the first 

 to the second place ; if this should be unknown, it would certainly be impossi 

 ble to distinguish the ascending from the descending node. 



After the angles Q,,i, are found, the arguments of the latitude u,u , will be 

 obtained by the formulas, 



cos t cos t 



which are to be taken in the first or second semicircle, according as the corre 

 sponding latitudes are north or south. To these formulas we add the following, 

 one or the other of which can, at pleasure, be used for proving the calculation : 



cos u = cos /3 cos (X 8 ), cos rf = cos /? cos (X Q ), 



sinS . , sinj? 



smtt = -^., smw = -T J T, 



cvr\ cin t 



COS t COS t 



