, * 



4 a 3 A *i* A * * ^ 



102 RELATIONS BETWEEN SEVERAL [BOOK I. 



t \ TJ\ a cos (5 4) b 



psin.(A P) = -- AT-JS- A^~ 



* ^ sin (B A) 



p cos (-4 P) = a. 



For ff= B, they acquire a similar form ; but for TT= $ ( A -(- B} they become 



so that the auxiliary angle t being introduced, of which the tangent = ^, it 



becomes 



tan(M + i# P) = tan(C 45) cotan i(# 4). 



Finally, if we desire to determine p immediately from a and b without previ 

 ous computation of the angle P, we have the formula 



p sin (B A) v/ (aa -\- bb 2 ab cos (B A)), 

 as well in the present problem as in II. 



79. 



For the complete determination of the conic section in its plane, three things 

 are required, the place of the perihelion, the eccentricity, and the semi-parameter. 

 If these are to be deduced from given quantities depending upon them, there 

 must be data enough to be able to form three equations independent of each 

 other. Any radius vector whatever given in magnitude and position furnishes 

 one equation : wherefore, three radii vectores given in magnitude and position are 

 requisite for the determination of an orbit ; but if two only are had, either one 

 of the elements themselves must be given, or at all events some other quantity, 

 with which to form the third equation. Thence arises a variety of problems 

 which we will now investigate in succession. 



Let r, /, be two radii vectores which make, with a right line drawn at pleasure 

 from the sun in the plane of the orbit, the angles N, N , in the direction of the 

 motion ; further, let IT be the angle which the radius vector at perihelion makes 

 with the same straight line, so that the true anomalies N IT, N IT may 

 answer to the radii vectores r, r ; lastly, let e be the eccentricity, and p the semi- 

 parameter. Then we have the equations 



