20 



RELATIONS PERTAINING SIMPLY 



[BOOK I. 



18. 



In the PARABOLA, the eccentric anomaly, the mean anomaly, and the mean 

 motion, become = ; here therefore these ideas cannot aid in the comparison of 

 the motion with the time. In the parabola, however, there is no necessity for an 

 auxiliary angle in integrating r r d v ; for we have 



and thus, 



frrdv = i pp (tan $ v -j- tan 3 if)-)- Constant. 



If the time is supposed to commence with the perihelion passage, the Constant 

 = ; therefore we have 



by means of which formula, t may be derived from v, and v from t, when p and 

 jit are known. In the parabolic elements it is usual, instead of p, to make use of 

 the radius vector at the perihelion, which is | p, and to neglect entirely the mass 

 [&amp;gt;. It will scarcely ever be possible to determine the mass of a body, the orbit of 

 which is computed as a parabola ; and indeed all comets appear, according to the 

 best and most recent observations, to have so little density and mass, that the 

 latter can be considered insensible and be safely neglected. 



19. 



The solution of the problem, from the true anomaly to find the time, and, in 

 a still greater degree, the solution of the inverse problem, can be greatly abbrevi 

 ated by means of an auxiliary table, such as is found in many astronomical works. 



