SECT. 3.] PLACES IN ORBIT. 139 



equations 13, 13* assume the form, 



and so, are wholly identical with those at which we arrived in the ellipse (15, 15*, 

 article 91). Hence, therefore, so far as h or H can be considered as known, y or 

 Y can be deduced, and afterwards we shall have 



1-1 -! 7 



[iv] = , 



l - JT 



From these we gather, that all the operations directed above for the ellipse serve 

 equally for the hyperbola, up to the period when y or Y shall have been deduced 

 from h or H; but after that, the quantity 



mm , -,- MM 

 -y~y- -TT&amp;gt; 



which, in the ellipse, should become positive, and in the parabola, 0, must in the 

 hyperbola become negative : the nature of the conic section will be defined by 

 this criterion. Our table will give C from z thus found, hence will arise the cor 

 rected value of h or H, with which the calculation is to be repeated until all 

 parts exactly agree. 



After the true value of s is found, c might be derived from it by means of the 

 formula 



but it is preferable, for subsequent uses, to introduce also the auxiliary angle n, 

 to be determined by the equation 



hence we have 



c = tan 2 n + y/ (1 + tan 2 2 n) = tan (45 + n). 



