ON THE REDUCTION OF THE GENERAL EQUATION. 289 



two values differing by 180° which, as before noted, only changes the 

 sign of p, and gives for each value the same line), which are also 

 parallel, and to each of which corresponds a single focus, given by 

 the corresponding values of li, k, from equations (4) and (5)*. 



These values of p may however in particular cases either coincide, 

 or be both imaginary, or one or both may be infinite or indeterminate : 

 it will however be more simple to deduce from our equations the 

 ordinary constants of the curve, which may be effected as follows : 



The equation to a directrix being 



x cos 6 -f y sin — jp=0, 

 that to a line drawn through the corresponding focus at right angles 

 to the directrix will be 



x — h y — k 

 coa siu 6 



The length of the perpendicular dropped on th.h from the origin is 



h sin 8 — h cos 6 

 which by virtue of (4) and (5) is equal to 



A. (d sin 6 — e cos 0) 



or, denoting this by K, 



i-r ,a+b—m , 7 . ,, As 



K — i — r— (d sin — e cos 6). 



This expression being the same whichever focus be taken, it 

 follows that the line thus determined (the ' transverse axis') passes 

 through both foci and is at right angles to both directrices ; and, from 

 the mode of generation, the curve must be symmetrical with regard 

 to it. 



The curve is also plainly symmetrical with regard to a line parallel 

 to both directrices and midway between them : the length of the 

 perpendicular dropped from the origin upon this line (the 'conjugate 

 axis') is the semi-sum of the values of p : calling this H we have 

 from the equation for p, 



H = — a (d cos + e sin 6) 



a + b + m , 



= — | -^zi^T ( d cos0 + e sm 0)> 

 Projecting H and K upon the axes of x and y successively, we 

 obtain the co-ordinates (#,' y") of the intersection of these two lines 



* These values arc as follows 



p ==- |. ? + b + m \— (dcoa + csin e) ■ \ JL{fuH +b& — 2ode — ob—c».f) ) U 

 ah — c* ' m -"J 



A= 1 — \-(bd — ce)i.\ h(»-a + 4) (ae2 + bd.2 - 2 cdo - ab -c2.f) I *] 



ab-& L < > J 



and a similar expression for k by interchanging a and b,d and e. A discussion of thom would 

 lead to the same results obtained more simply in tho text. 



V 



