554 
PROBLEMS AND SOLUTIONS. 
[485 
I have obtained for Prob. 2 a solution which I believe to be accurate; viz. the 
number of the conics (4, 1), (that is, the conics which have with a given curve a 
5-pointic intersection and also a 2-pointic intersection, or ordinary contact), is 
= 10 v? +10 nm — 20 m 2 — 130 n + 140 m 4-10 k (m + n — 9) — 4 [(?i — 3) k + (m — 3) t] 
where i (the number of inflexions) is = 3n — 3m + k, but I prefer to retain the fore 
going form, without effecting the substitution. 
[Vol. v. pp. 88, 89.] 
1890. (Proposed by Professor Cayley.)—Find the equation of a conic passing 
through three given points and having double contact with a given conic. 
Solution by the Proposer. 
Let the given points be the angles of the triangle (x = 0, y — 0, z = 0), and let 
the equation of the given conic be U = (a, b, c, f, g, li§x, y, z) 2 = 0; then the equation 
of the required conic is 
TJ— (x \fa + y \/5 + z \fc) 2 = 0, 
for this is a conic having double contact with the conic U= 0, and, since the terms 
in (¿c 2 , y 2 , z 2 ) each vanish, it is also a conic passing through the given points. 
It is clear that there are four conics satisfying the conditions of the Problem, 
viz. putting for shortness 
P =x*Ja + y*Jb+z \/c, P 1 = — x*Ja + y*Jb+z \/c, 
P 2 = x \ja — y \/b + z Vc, P 3 = x *Ja + y \Jb — z Vc, 
the four conics are 
U-P 2 = 0, U-P 2 = 0, U-P.? = 0, U- P : r = 0. 
It may be remarked that the conics P, P l have a fourth intersection lying on the 
line y\]b + z\jc = 0, and the conics P 2 , P 3 a fourth intersection lying on the line 
y\jb — z\!c\ which two lines are harmonics in regard to the lines y = 0, z — 0. 
Similarly the conics P 1; P 2 have a fourth intersection on the line x \!a + z \]c — 0, 
and the conics P, P 3 a fourth intersection on the line x ^Ja — z \Jc = 0; which two lines 
are harmonics in regard to the lines z = 0, x = D. And the conics P 2 , P 3 have a fourth 
intersection on the line x \/ft + y \/b = 0, and the conics P, P 2 a fourth intersection on 
the line x^a — y^/b = 0; which two lines are harmonics in regard to the lines 
x — 0, y = 0. It may further be remarked that the equations of any two of the four 
conics may be taken to be 
ayz + fizx + yxy = 0, oiyz + fi'zx + yxy = 0.