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PROBLEMS AND SOLUTIONS. 
[485 
and thence 
f{-fW - P'1) + 9 (7* - 7'«) + h («P - olp)) = (- 2be), 
and the equations become 
x : y : z = a {BC + B'G) : b(CA' + C'A) : c(AB' + A'B), 
where BC + B'C, CA' + C'A, AB' + A'B, substituting therein for f, g, h the values 
. a _. - , —r-—r-, —775, are respectively functions of the fourth degree in a, b, c: 
J3y -/3y yet'-y'a’ ol(3-o'/3 u j o ’ ’ 
hence (a, b, c) being connected by the relation a + b + c = 0, x, y, z are proportional 
to quintic functions of (a, b, c), or what is the same thing, writing a, b, c = l, 6,—1 — 6, 
then x, y, z are proportional to quintic functions of 6, that is, the locus is a unicursal 
quintic curve. 
That the curve passes through the points (a', /3', y') and (a, /3, y) appears by con 
sidering the conics fyz + gzx + hxy = 0, which pass through these points respectively. 
For the first of these conics we have f:g : h = a (/3y — /3'y) : /3(ya'-y'a) : a(/3y'—/3'y) 5 
the equation 
A’- + B ‘ -A , + o A ,a = o, 
py — /3 y ya — y a a/3 — a /3 
reduces itself to x (/3y' — /3'y) + y (y a' — y'a) + 0 (a/3' — a'/3) = 0, and as the other equation 
A' 2 
ßy - ß'y 
+ B' 2 
y 
ya — y a 
-+C 2 
y a — y a 
71=0, 
is that of a line through (a', /3', y') the two lines meet of course in the point 
(of, /3', y'). And the like for the conic 
/ : g : h = a! (/3y' - /3'y) : /3' (ya' - y'a) : y' (a/3' - a'/3). 
If the triangle is equilateral, and (x, y, 2) are respectively proportional to the 
perpendicular distances from the three sides, then we have for the circular points at 
infinity 
( a > /3, y) = (1) a>, (o 2 ), (a, /3', y') = (l, ® 3 , co), 
where co is an imaginary cube root of unity. These values give 
/3y' — /3'y = ya' — y'a = a/3' — a'/3 = co 2 — co 
aa' = (3/3' = yy' = 1, By' + /3'y = ya' + y'a = a/3' + a'/3 = — 1; 
and the expressions for (x, y, z) take the form 
x : y : 2 = a {26 2 c 2 — a 2 (a 2 + 5 2 + c 2 )} 
: 5 {2c 2 a 2 — b 2 (a 2 + 5 2 + c 2 )} 
: c {2a 2 5 2 — c 2 (a 2 + 5 2 + c 2 )}, 
or, what is the same thing, reducing by means of the relation a + b + c = 0, 
x : y : z = ci(a 4 — 2a?bc - 2b 2 c 2 ) : b ( b 4 — 2b 2 ca — 2c 2 a 2 ) : c (c 4 — 2c 2 ab — 2a 2 b 2 ),