ON THE BEUTjCTION OF THE GENERAL EQUATION. 293 



= y {ab— c») 



and therefore 



i « 1 



ez_6/ d--af 



c B +6 9 c 2 h«» 



In this case the curve reduces to two parallel straight lines, paral- 

 lel to and equidistant from the transverse axis (which still remains 

 determinate in position), the distance between them being double the 

 foregoing value of B. 



If e 2 =bf (which is the same as <i 2 = of), these two lines coalesce 

 into the transverse axis, and if e'—bf be negative, they are imagi- 

 nary. 



In general, however, for the parabola, the elements obtained in the 

 ellipse and hyperbola are insufficient when the co-ordinates of the 

 centre become infinite : the original equations ( I ) (6) admit how- 

 ever in this case of easy solution. For, since ab — <? a =0, we have 



m=a + b, e"=l, a— — -, and the equations become 

 a+b 



sin 2 i9— — r , sin 9 cos 6 = — r 

 a + b ' a + b 



—\d=]i—2> cos $ (4) 



—Xe=Jc—p sin (5) 



Xf=]r + 7, 2 - 2 ^ (6) 



from which we obtain at once by simple equations 



1 d*+ei _(a + 6)_/ 



P " -2 (a + b) dco&d+e sia 



1 c(c*-d*)+2ade-(a+b)cf 



h = 



* = 



2(a + 6) cd -« e 



1 c(di-a )+2bde ~{a+b)cf 



\{a + b) ec-bd 



If we draw a line through the focus parallel to the directrix, the 

 portion intercepted by the curve is double the distance of the focus 

 from the directrix, as is evident from the mode of generation. 

 If we call this portion L (the 'latus rectum'), we have 

 \ L — h cos 6 +k sin 6 —p 



= \(d cos 6 + e sin 6), by (4) nod (5). 



Henco 



1 abe^ + b^d" 1 -Ibcde — \ (bd-Ce) 



(a+b)" 1 ab +b* (a+b)' c» + 6'J 



