134 RELATIONS BETWEEN SEVERAL [BOOK I. 



Hence, cos F being eliminated, 



2r/ 

 LJ .?&amp;gt; --&amp;gt;__/_ 



If, accordingly, we adopt here also the equations 9, 9*, article 88, the first for 

 cos/ positive, the second for cos/ negative, we shall have, 



r7*i . - 1 / v r/ 



L J P--2Lco S f&amp;gt; 



which values being substituted in equation 3, preserving the symbols m,M, with 

 the meaning established by the equations 11, 11*, article 88, there result 



[8] w =/ 



[8*] M= i*-f |Z . 



These equations agree with 12, 12*, article 88, if we there put g = 0. Hence it is 

 concluded that, if two heliocentric places which are satisfied by the parabola, are 

 treated as if the orbit were elliptic, it must follow directly from the application 

 of the rules of article 19, that x== 0; and vice versa, it is readily seen that, if 

 by these rules we have x = 0, the orbit must come out a parabola instead of 

 an ellipse, since by equations 1, 16, 17, 19,20 we should have = oo, a=&amp;lt;x&amp;gt;, 

 (f = 90. After this, the determination of the elements is easily effected. Instead 

 of p, either equation 7 of the present article, or equation 18 of article 95 f might 

 be employed : but for F we have from equations 1, 2, of this article 



tan J J^= ? ~*^ cotan / = sin 2 w cotan i /, 



if the auxiliary angle is taken with the same meaning as in article 89. 



We further observe just here, that if in equation 3 we substitute instead of 

 p its value from 6, we obtain the well-known equation 



kt = H r + &amp;gt; + cos/, y/r /) (r -(- / 2 cos/, y/r/ )* ^ 2. 



t Whence it is at once evident that y and J&quot; express the same ratios in the parabola as in the 

 ellipse. See article 95. 



