SECT. 3.] PLACES IN ORBIT. 107 



tion with the third gives rise to the following mode of proceeding. The auxil 

 iary angle may be determined by the equation 



_ - 



/ &mi(N N) 



r&quot; 



and we shall have 



tan (* N+ IN +IN&quot; II} = tan (45 -f Q tan 



Two other solutions wholly analogous to this will result from changing the second 

 place with the first or third. Since the formulas for - become more complicated 

 by the use of this method, it will be better to deduce e and p, by the method of 

 article 80, from two of the equations (I.). The uncertainty in the determination 

 of IT by the tangent of the angle J JV-f- i N -(- J N&quot; -IT must be so decided 

 that e may become a positive quantity : for it is manifest that if values 180 dif 

 ferent were taken for 77, opposite values would result for e. The sign of p, how 

 ever, is free from this uncertainty, and the value of p cannot become negative, 

 unless the three given points lie in the part of the hyperbola away from the sun, 

 a case contrary to the laws of nature which we do not consider in this place. 



That which, after the more difficult substitutions, would arise from the appli 

 cation of the first method in article 78, II., can be more conveniently obtained in 

 the present case in the following manner. Let the first of equations II. be multi 

 plied by cos 4 (N&quot; N \ the third by cos 4 (N r - N), and let the product of 

 the latter be subtracted from the former. Then, lemma I. of article 78 being 

 properly applied,* will follow the equation 



cotan 





&quot; N } 4 (; 7) cotan * (N N) 



By combining which with the second of equations H 77 and - will be found ; thus, 

 77 by the formula 



Putting, that is, in the second formula. A = %(N&quot;N ), B=%N-\-%N&quot; 77, C=$(NN ). 



