SECT. 1.] TO POSITION IN THE ORBIT. 19 



which, nevertheless, is not a maximum here, since the difference between v and 

 E may still increase beyond 9. This last difference becomes a maximum for 

 d (f E ) = or for d v = d E, where the eccentricity is clearly to .be regarded 

 as constant. With this assumption, since in general 



dv AE 



sinu ~ sin.fi 



it is evident that we should have sin v = sin E at that point where the difference 

 between v and E is a maximum ; whence we have by equations VIII., III., 



r = a cosy, e cosE = 1 cos 9, or cos E = -(- tan J 9. 

 In like manner cos v = tan 9 is found, for which reason it will follow * that 



v = 90 -{- arc sin tan $ 9, E = 90 arc sin tan i 9 ; 

 hence again 



sin E V (1 tan 2 } 9) = * 



cos qp 

 so that the whole equation of the centre at this point becomes 



2 arc sin tan i 9 -|- 2 sin i 9 y cos 9, 



the second term being expressed in degrees, etc. At that point, finally, where 

 the whole equation of the centre is a maximum, we must have d v = d M, and 

 so according to article 15, r =. a \J 0039 ; hence we have 



POSI&amp;gt;- 1 cos*? . E._l V/cosg)_ 1 cos go tan 9 



l^Uo t/ - OLIO _L/ - 7^ j 7 -^ - i j I . 



e e e (1 -\- y cos &amp;lt;f) l-|-v cos &amp;lt;p 



by which formula E can be determined with the greatest accuracy. E being 

 found, we shall have, by equations X., XII., 



equation of the centre = 2 arc sin !1!L2_2_!1 _|_ e sm J g r . 



y cosgj 



We do not delay here for an expression of the greatest equation of the centre by 

 means of a series proceeding according to the powers of the eccentricities, which 

 several authors have given. As an example, we annex a view of the three 

 maxima which we have been considering, for Juno, of which the eccentricity, 

 according to the latest elements, is assumed = 0.2554996. 



* It is not necessary to consider those maxima which lie between the aphelion and perihelion, 

 because they evidently differ in the signs only from those which are situated between the perihelion and 

 aphelion. 



