SECT. 1.] TO POSITION IN THE ORBIT. 23 



r== 



2 e cos (v ifj) cos (v -f- i/&amp;gt;) &quot; 



For # = 0, the factors cos $ (v tp), and cos (y -)- y), in the denominator of this 

 fraction become equal, the second vanishes for the greatest positive value of v, 

 and the first for the greatest negative value. Putting, therefore, 



cos ^ (v -(- if&amp;gt;) ~ 



we shall have u = 1 in perihelion ; it will increase to infinity as v approaches its 

 limit 180 i//; on the other hand it will decrease indefinitely as v is supposed 

 to return to its other limit (180 1/&amp;gt;) ; so that reciprocal values of u, or, what 

 amounts to the same thing, values whose logarithms are complementary, corre 

 spond to opposite values of v. 



This quotient u is very conveniently used in the hyperbola as an auxiliary 

 quantity ; the angle, the tangent of which is 



/e 1 



can be made to render the same service with almost equal elegance ; and in order 

 to preserve the analogy with the ellipse, we will denote this angle by I F. In 

 this way the following relations between the quantities v, r, u, F are easily brought 

 together, in which we put a = b, so that b becomes a positive quantity. 



I. l=.p cotan 2 y 

 H. r = p - = _ pcoay _ 



1 -}- e cos v 2 cos J (v y) cos (v -\- 1/&amp;gt;) 



HI. 



_ t ,45 

 - 



y 1 _ i / I 1 -. _ 1 -)- cos if&amp;gt; cos v _ e -f- cos v 



cosl ~ i u 2 cos ^ (v 1/&amp;gt;) cos^ (v-\-\f&amp;gt;) l-f-ecos* 



By subtracting 1 from both sides of equation V. we get, 

 VI. smJ, = 



