Foci of a Conic Section. 441 
taking into account the directrix, does, notwithstanding, ad- 
mit of a short and simple solution; and I think the investi- 
gation here given will not be without interest to the mathe- 
matical readers of the Magazine,—the discussion of the pro- 
blem under this aspect leading to a series of equations very 
similar to those which occurred in my own form of the pro- 
blem. 
Prop.—A point can always be found in the plane of a line 
of the second order referred to any angle of ordination, such 
that its distance from any point of the curve is a rational linear 
Junction of the coordinates of that point of the curve. 
Let ay? + bvy+eu?+dy+exr+f=0 be the curve; 
a the angle of ordination; 4 & the point to be found: then it 
is affirmed that 4, k, p, g, r can be so determined that for all 
corresponding values of w and y we shall have 
(y — ky +2 (y — 2) (@— A) cosa + (v9 — BP 
of the form (py + gz 4+ 71r)*. 
For multiply the equation of the curve by; then we shall 
have an identity between the two following expressions :— 
Nay? +rbey + rca? +rdy + rea + Af, and 
(y — k)? +2 (y—k) (a —h) cosa + (vw —h)?— (py t+ gz +7)’. 
Equating the homologous coefficients of these expressions, 
we get 
Rie et nes cs oe A daly Se vem alte it} ols) 
RO 2s IY oe ee mr oe earths 1 Oe) 
FO SE eee he ca Bi Bs shee st Si Oe 
r~Ad=—2(k+hcosatpr) .... « (4) 
Ae=—2(R+kcosa+qr) . + +. « - (5) 
f=? +2hk cosa + kh? —9 (6.) 
We have now to show how the values of h, k, p, 5 75 A 
may be determined, and to prove that they are real. 
From (1, 2, 3.) we get at once 
pe=1—Aa, 2pq = 2cosa —Abd, and g? = 1 — Ac, 
Whence 
4(1 —Aa) (1 —Ac) = 4p? q? = (2p q)? = (2 cosa — Ab); 
or collecting and arranging the terms in x we have 
4 sin? a. 4; — 4 (a — Beosa + ¢).— = 0? — 4a. (7.) 
Now, putting as in my former paper, page 26, Q and R for 
their values, we have 
1 Q+R 2(R—Q). 
ES aaa OA Rm pace theo mng PKB 
and 4 is thereciprocal of the u? in the solution before referred to, 
Using the same notation as before, we get from (1, 3.), 
