SECT. 1.] TO POSITION IN THE ORBIT. 9 



8. 



Let us now collect together those relations between the anomalies and the 

 radius vector which deserve particular attention, the derivation of which will 

 present no difficulties to any one moderately skilled in trigonometrical analysis. 

 Greater elegance is attained in most of these formulas by introducing in the 

 place of e the angle the sine of which = e. This angle being denoted by &amp;lt;p, we 

 have 



_ ee ) cosy, y/(l + e) = cos (45 i 9) y/2, 



e) = 2 cosy, \/(l-\-e) y/(l e) = 2 sin i &amp;lt;p. 



The following are the principal relations between a, p, r, e, (f, v, E, M. 

 I. p a cos 2 y 

 II. r = TJ f 



i -\-e cos w 



III. r = a(l ecosE) 



j-y cos v -j- e cos ^J e 



1 -[- e cos w 1 e cos ^? 



V. siniJi r =\/ HI cos^&quot;) =sin^i/r 



e cos v 



ini^^VHl cos J?) =sinif YT 



V 1 -\-ecos 



P 



VL cosi^= v/i (1 -j- 



= sm 



e cos 



VII. tan iJ?= tan i tan (45 



VIII. sin.E ^: 



r sm v cos qo r sin u 



p a cos qj 



IX. r cos = a (cos E e) = 2 a cos ( * E + } 9 + 45) cos ( j i 9 -- 45) 



X. sin i (y ^/) = sin J y sin v J - = sin J y sin _&quot; t/ 



XI. sin i(y-(-^ )^cos^9sin^i -=r cos 

 XII. M=E 





