RELATIONS BETWEEN SEVERAL [BOOK I. 



results 0.4159824 ; lastly, it is deduced from formula VUL, 0.4051103 : the two 



last values differ so much from the truth that they cannot even be used as ap 

 proximations. 



The exposition of the second method will afford an opportunity for treating 

 fully a great many new and elegant relations ; which, as they assume different 

 forms in the different kinds of conic sections, it will be proper to treat separately ; 

 we will begin with the ELLIPSR 



Let the eccentric anomalies E, E , and the radii vectores r, r, correspond to 

 two places of the true anomaly v, v , (of which v is first in time) ; let also p 

 be the semi-parameter, e = sin (p the eccentricity, a the semi-axis major, t the 

 time in which the motion from the first place to the second is completed ; finally 

 let us put 



Then, the following equations are easily deduced from the combination of for 

 mulas V., VI., article 8 : 



[1] b smff = sin/, ^rr , 



[2] l)M\G = $m.F.\jrr , 

 p cosy = (cos i v cos i v . (1 -4- e) -\- sin k v sin i v . (1 e}~) y/ r r, or 



[3] p cosg = (cos/ -4- e cos F) y/ r r, and in the same way, 



[4] p cos = (cos F-\- ecosf) \Jrr . 

 From the combination of the equations 3 and 4 arise, 



[5] cos/, y/r/ (cosy e cos G) a, 



[6] cos I 1 . \Jrr = (cos G ecosg)a. 

 From formula III., article 8, we obtain 



[7] r r = 2 a e sin g sin G, 



r -\-r = 2 a 2aecosg cos G = 2asin 2 y-|- 2 cos/cosy y/rr ; 

 whence, 



2 cos /cos g-jrr 



~ 



