204 DETERMINATION OF AN ORBIT FROM [BOOK II. 



equator). All this may be effected by the solution of a single spherical tri 

 angle. Let 8 be the longitude of the ascending node ; i the inclination of the 

 orbit ; g and g&quot; the arguments of the latitude in the first and third observations ; 

 lastly, let I & = h, I&quot; - - Q, = li . Calling, in figure 4, & the ascending node, 

 the sides of the triangle Q, AC will be AD c, g, h, and the angles opposite to 

 them, respectively, i, 180 y, u. We shall have, then, 



sin i i sin i (g -\- h] = sin i (A!? t) sin J (y -j- u) 

 sin J i cos (g -\- h) = cos i (AD ) sin i (y u) 

 cos i z sin k (g h} = sin i ( AZ/ ) cos I (y -(- w) 

 cos J z cos % (y h} = cos i (AZ/ ) cos (y ?;). 



The two first equations will give i (#-|-A) and sin ^ the remaining two i (y Ji) 

 and cos H; fromy will be known the place of the perihelion with regard to the 

 ascending node, from h the place of the node in the ecliptic ; finally, i will be 

 come known, the sine and the cosine mutually verifying each other. We can 

 arrive at the same object by the help of the triangle &A&quot;C , in which it is only 

 necessary to change in the preceding formulas the symbols g, h, A, L, y, u into y&quot;, 

 h&quot;, A&quot;, &quot;, y&quot;, u&quot;. That still another verification may be provided for the whole 

 work, it will not be unserviceable to perform the calculation in both ways ; 

 when, if any very slight discrepancies should show themselves between the values 

 of i, Q, , and the longitude of the perihelion in the orbit, it will be proper to take 

 mean values. These differences rarely amount to OM or .2, provided all the 

 computations have been carefully made with seven places of decimals. 



When the equator is taken as the fundamental plane instead of the ecliptic, 

 it will make no difference in the computation, except that in place of the points 

 A, A&quot; the intersections of the equator with the great circles AB, A B&quot; are to be 

 adopted. 



