238 OUTLINES OF NATURAL PHILOSOPHY. 



j 

 From ( 236, b.) b c = r v ; now, c =? ! ; therefore 



a 



bdf*=ra%v,or b*d*f = av*r*; and because 

 b* = AS X SB = r(2a r), 

 r (2 a V) d/= a w r% and 



Thus the semi- transverse axis is found, and from it, 

 with the focus S, and the apsis A, the conic sec- 

 tion may be described. 



b. The conic section will be a circle, when a = r, or, 



d* f 

 when d*f = w* r, or w* = -^-. If is such that 



2 d* f 

 2 d?/* = w* r 9 or v* =: -^, the denominator is 0, 



and the value of a becomes infinite, so that the tra- 

 jectory is a parabola, of which the focus is 5, and 

 the parameter 4- r. 



c. When 2 ^V> v * r, the value of a is affirmative, and 

 the conic section is an ellipsis ; and this ellipsis has 



d* f 

 its higher apsis at A, if v* ^^ -4- ; but when w* 



d* f 2 d* f 



is between the limits of and - ^-, the lower 

 r r 



apsis is at A. 



d. When 



