TRANSACTIONS OF SECTION A. 



481 



comparing with u = o we get 



1 — e* cos'^ 6 1 — e^ sin' 6 — e^ sin 6 cos d eh) cos 6 — a 



gi + ff^- eV _ 1 



= -^say 



a a 



f 

 Eliminating e and 6 we get 



X-i -(a + b)\ + ab -h"- = (A) 



Since xy are the coordinates of any point on a directrix, by eliminating p, 6, a^, e 

 from .X- cos + y sin ^ = ^> and the above equalities, we shall get the equation to the 

 directrices 



The result of the elimination is 



(IT- ©'-"•' w 



Kdy 



I do not exhibit the work, because a quicker method of obtaining it will be given 

 in the note. 



Using (A), (B) may be shown to represent two parallel straight lines. 



Thus one value of \ from (A) gives the real directrices, and the other the 

 imaginary. 



I find further that B may be resolved into 



/ , du , du, n , — 



ahg 



where A = hhf 



(0) 



and the sign between the radicals on the left side is that of j—- 

 It follows that 



vxr^ 



du 



dx 



, du n 



is the equation to an axis of the conic. 

 In virtue of (A) this is eqiuvalent to 



/, ,^ du 7 dii 

 (X _ 6) + h -- 



ax dy 



, du ,v . du 



h— + (X — «) 



or 



dy 



= 







(D) 



(E) 



d.v ^ ^ dy 



Having obtained the equations to an axis and to a dii-ectrix we can obtain the 

 equation to « latus rectimi (the polar of their intersection). Using Dr. Salmon's 

 notation for the reciprocal coefficients the result is 



N/X^6(C,r - G) + -/)r^(Cy - E) = ± (a + b - 2X) ^/a^ . . . . (F) 



The quantity a + b - 2X may be shown to be the expression denoted by R in 

 Dr. Salmon's conies (Ex. 3, Art 157, and elsewhere). 



Note. 



The form of tlie equation to the director circle of m = o, viz. with Dr. Salmon's 

 notation, 



v=C (.r- + r) - 2G.r-2Fy + A +B = 

 shows that the straight lines joining any point on it to the circular points at infinity 

 are conjugate with respect to the conic. 



This is a particular case of the following theorem : — If from two fixed points 

 in the plane of a conic straight lines be drawn conjugate with respect to the 

 conic, the locus of their intersection is, in general, a conic passing through the 

 two given points. Also since a tangent is conjugate to anv straight line passing 



1880. I I 



