294 ON THE REDUCTION OF THE GENERAL EQUATION. 



We will now proceed to recapitulate the values of the elements 

 necessary and sufficient for the determination of the curve in the 

 general cases. 



For the ellipse and hyperbola, ihe co-ordinates of the centre are 



ce—bd cd—ae 



ab-c* ' J ab-c'i ' 



the semi-axes, transverse and conjugate, are given by the values 



a (a6-cV \ \ 



In the ellipse ab—cr is positive, and m is always to be taken pos- 

 itive : in the hyperbola ab—c z is negative, and in must be taken of 



the same sign as the quantity within the \ — [• . 



The inclination (0) of the transverse axis to the axis of x is then 

 given without ambiguity by the equation*?. 



2 cos z $ =1 — - 11 -. sin 6 cos 0= — . 



m m 



being measured by revolution from the positive part of the axis of 



x to that of y. 



In the parabola, ah — c*=0 ; the co-ordinates of the focus are 



1 -d a )+2a<fe-(g+6)c/ 



h = 

 k = 



'l{a+b) cd—ae 



1 c(d?-<?)+2bde-(a+b) cf 



2(a+b) ce-bd 



The position of the directrix is given by the angle made by its 

 normal with the positive part of the axis of x (& being measured by 

 revolution towards that of y) and the length p of this normal, includ- 

 ing sign as indicating a direct or backward measurement from the 

 origin. These are given without ambiguity by the equations 



Bin* $ = — - , sin cos = — - , 

 a+b ' a+b ' 



1 d*+e* — (a+b) f 



* 2 (a+b) ' d cos 0+esin 6 



These elements are sufficient to determine the position and dimen- 

 sions of the curve as well as the direction towards which its concavity 

 is turned ;* but the latus-rectum L is also given directly by the value 

 _ _i_ (bd-ceY 

 * — (a+by~ • 68 +c s ' 

 In particular cases, the ellipse may degenerate into a point, or be 

 wholly imaginary; the hyperbola may degenerate into two intersecting 



• In the ardimrj methods of reduction, tliis direction ih undetermined. 



