146 RELATIONS BETWEEN SEVERAL [BOOK 1. 



Now, if we denote the chord by 9, we shall evidently have 



Q () = (r cos if r cos v) 2 -\- (r sin v r sin v) z , 

 and, therefore, by equations VIII., IX., article 8, 



Q (&amp;gt; = a a (cos E cos Ef -\- a a cos 2 y (sin E sin E) z 



= 4 a a sirfg (sin 2 G -(- cos 2 (p cos 2 G) = 4 a a sin 2 ^ (1 e e cos 2 G). 



We introduce the auxiliary angle h such, that cos h = e cos G ; at the same time, 

 that all ambiguity may be removed, we suppose h to be taken between 0-and 

 180, whence sin h will be a positive quantity. Therefore, as g lies between the 

 same limits (for if 2y should amount to 360 or more, the motion would attain to, 

 or would surpass an entire revolution about the sun), it readily follows from the 

 preceding equation that ^ = 2 smg sin A, if the chord is considered a positive 

 quantity. Since, moreover, we have 



r-\-r = 2(1 ecos^cos^) = 2a(l cosy cos h), 

 it is evident that, if we put h g = #, h -\-g = t., we have, 



[1] r -f r &amp;lt;j = 2 a (1 cos 8} = 4 a sin 2 } d, 

 [2] r-j-/-|-9 = 2o(l cos e) = 4 a sin 2 i t- . 



Finally, we have 



3 3 



Itt = a 7 (2^ 2 &amp;lt;?siny cos 6 ! ) = - (2^ 2 siny cos A), 

 or 



^ / 

 [3] =:a*(a sine 



Therefore, the angles d and e can be determined by equations 1, 2, from 

 &amp;gt; -)- ^&quot; ? ?&amp;gt; and a ; wherefore, the time t will be determined, from the same equa 

 tions, by equation 3. If it is preferred, this formula can be expressed thus : 



,, f/ 



k t = a ( 



\ 



2a 



2a(r + r )o 2 a (r-\-r ) Q 



arc cos - - sm arc cos - 



2a 2a 



_ p . 



- arc cos - 4- sm arc cos - 



2a 2a 



But an uncertainty remains in the determination of the angles &amp;lt;?,e, by their 

 cosines, which must be examined more closely. It appears at once, that d 

 must lie between 180 and + 180, and e between and 360 : but thus 



