142 RELATIONS BETWEEN SEVERAL [BOOK. I. 



2 (I z) cos/, v/r/ _ 2mmcos/. y/r/_ 



2 ~ 



klctt 



_ _ 



tan 2 2 7i y#tan 2 2n ~4yy rr cos 2 /tan a 2n 



kktt 



2(-|-z)cos/.v/r/ _ - 2 MMcosf. v/ r / _ _ _ 

 tan a 2n ~&quot;FTtan 2 2n ~ 4 T Tr / cos 2 / tan 2 2 n 



_ sin/.tan/.y/r/ _ yy sin/, tan/, y/r/ _ _ /yr/sin2/\ 2 

 * : 2(/ z) 27WOT ~\ /!; / 



- sin /- 1? 11 /: V^ _ rrsin/.tan/.y/r/ _ / TV/sinj2/\2 

 &quot;~ ~V /fci! / 



The third and sixth expressions for p, which are wholly identical with the form 

 ulas 18, 18*, article 95, show that what is there said concerning the meaning 

 of the quantities y, Y, holds good also for the hyperbola. 



From the combination of the equations 6, 9, article 99, is derived 



by introducing therefore y and w, and by putting (7= tan (45 -j-^V), we have 



[20] tan2^= 2si 7 ton 9 2(U . 

 sm/cos 2u 



C being hence found, the values of the quantity expressed by M in article 21, will 

 be had for both places ; after that, we have by equation III., article 21, 



G c 



tan J v = 



, 



tan j v = 



f-fj-. r- j 



(O-\-c) tan 

 Oc 1 



or, by introducing for C, c, the angles N, n, 



rnn 



= 



[22] tan^=- 



cos (iv n) tan ^ i/; 



Hence will be determined the true anomalies v, v , the difference of which com 

 pared with 2/ will serve at once for proving the calculation. 



Finally, the interval of time from the perihelion to the time corresponding to 

 the first place, is readily determined by formula XL, article 22, to be 



tan (45 

 tan (45 



