414] 
ON POLYZOMAL CUEVES. 
537 
the line of centres is the axis of the curve, but the centres A, B, G are not the foci, 
except in the case a" = 0, b" = 0, c" = 0, where the circles vanish. The condition for 
l, m, n is satisfied if l : m : n = (b — c) 2 : (c - a) 2 : (a — b) 2 ; these values, writing 
fl : fm : fn = b — c : c — a : a — b, give not only V/ + Vm + V?i = 0, but also 
afl + bfm + cfn = 0; these are the conditions for a branch containing (z 2 = 0) the 
line infinity twice; the equation 
(b — c) V(x — az) 2 + y 2 — a" 2 z 2 + (c — a) V(x — bz) 2 + y 2 — b" 2 z 2 + (a — b)\f(x — cz) 2 + y 2 — c" 2 z 2 = 0, 
is thus that of a conic, and if a” = 0, b" = 0, c" = 0, then the curve reduces itself to 
y 2 = 0, the axis twice. 
165. If il is not =0, then we have 
l : m : n = (/3 - 7) (/3' - 7') : (7 - a) (7' - <*■') : (a - ft) («' - /3')> 
viz., I, m, n are as the squared distances BC 2 , CA 2 , AB 2 , say as f 2 : g 2 : h 2 ; or when 
the centres of the given circles A, B, G are not in a line, then f g, h being the 
distances BG, CA, AB of these centres from each other, we have, touching each of 
the given circles twice, the single Cartesian 
/Vr +g *JW+h Vg 5 = 0, 
which, in the particular case where the radii a", b", c" are each = 0, becomes 
/V21 4-^V33 4-AV® =0, 
viz., this is the circle through the points A, B, G, say the circle ABG, twice. 
Article Nos. 166 to 169. Investigation of the Foci of a Conic represented by 
an Equation in Areal Coordinates. 
166. I premise as follows: Let A, B, G be any given points, and in regard to 
the triangle ABG let the areal coordinates of a current point P be u, v, w; that is, 
writing PBG, &c., for the areas of these triangles, take the coordinates to be 
u : v : to = PBG : PCA : PAB, 
or, what is the same thing in the rectangular coordinates (a, y, z= 1), if 
(a, a', 1), (6, b', 1), (c, c, 1), 
be the coordinates of A, B, C respectively, take 
u : v : tv = 
V, z 
x, y , z 
: 
X, y, z 
b, b\ 1 
c, d, 1 
a, a, 1 
c, c', 1 
a, a, 1 
b, b‘1 
C. VI. 
68