CT. 1.1 TO POSITION IN THE ORBIT. 



6. 



Let us proceed now to the comparison of the motion with the time. Putting, 

 as in Art. 1, the space described about the sun in the time t=$g, the mass of the 

 moving body = jit, that of the sun being taken = 1, we h&v&ff 

 The differential of the space = krrdv, from which there results 

 =frr&v, this integral being so taken that it will vanish for t = 0. This integra 

 tion must be treated differently for different kinds of conic sections, on which 

 account, we shall now consider each kind separately, beginning with the ELLIPSE. 



Since r is determined from v by means of a fraction, the denominator of which 

 consists of two terms, we will remove this inconvenience by the introduction of a 

 new quantity in the place of v. For this purpose we will put tan v ^ T = 



i -\-e 



tan % E, by which the last formula for r in the preceding article gives 



= 



n r\ 



^ r=r- ( 



Moreover we have ^ = y^, and consequently dv = f 



hence 



rrd( , == __.(l 



and integrating, 



e sin ^) ^Constant. 



(1 e ey 



Accordingly, if we place the beginning of the time at the perihelion passage, where 

 v = 0, E= 0, and thus constant = 0, we shall have, by reason of l ^_ ee = &amp;lt;*, 



In this equation the auxiliary angle E, which is called the eccentric anomaly, 

 must be expressed in parts of the radius. This angle, however, may be retained 



in degrees, etc., if e sin E and **V(H-f*) are a i so expressed in the same manner ; 



or 

 these quantities will be expressed in seconds of arc if they are multiplied by the 



