414] 
ON POLYZOMAL CURVES. 
575 
Hence, first writing a - x, b — x, c - x in place of a, b, c; then y 2 for 0, and 
(— a" 2 , — b" 2 , — c" 2 ) for (a, /3, 7); and finally introducing z for homogeneity, we find 
Norm {(b — c)^(x — az) 2 + y 2 — a" 2 z 2 + (c — a) V „ + (a — b) V „} = z 2 into 
s 2 ((b - c) 4 a" 4 + (c- a) 4 b" 4 + (a - b) 4 c" 4 
-2 (c - a) 2 (a - b) 2 b" 2 c" 2 -2 (a- b) 2 (b - c) 2 c" 2 a" 2 -2 (b - c) 2 (c - a) 2 a" 2 b" 2 ) 
— 4y 2 (b — c)(c— a)(a — b)[ (b — c) a' 2 + (c — a) b" 2 + (a — b) c" 2 ] 
— 4 (b — c)(c — a)(a — b){ (b — c) a" 2 (z 2 be — zx (b + c) + x 2 ) 
+ (c — a) b" 2 (z 2 ca — zx(c + a) + x 2 ) 
+ (a — b) c" 2 (z 2 ab — zx (a + b) + x 2 )) 
— 4>y (b — c) 2 (c — a) 2 (a — b) 2 , 
so that the equation (b — c) V2l° + (c — a) + (a — b) V® 0 = 0, in its rationalised form, 
contains (z 2 = 0) the line infinity twice, and the curve is thus a conic. If a" 2 = b" 2 = c" 2 = k" 2 , 
then the expression of the norm is 
= z 2 into — 4 (a — b) 2 (b — c) 2 (c — a) 2 (y 2 — k" 2 z 2 ), 
viz., when the three circles have each of them the same radius k", the curve is the 
pair of parallel lines y 2 — k" 2 z 2 = 0; and in particular when k" = 0, or the circles reduce 
themselves each to a point, then the curve is y 2 = 0, the axis twice. 
Annex IV. On the Trizomal Curves *JlU+Vm7 + \/nW =0, which have a Cusp, or 
two Nodes. 
The trizomal curve y/lU + + *JnW =0, has not in general any nodes or cusps: 
in the particular case where the zomal curves are circles, we have however seen how 
the ratios l : m : n may be determined so that the curve shall acquire a node, two 
nodes, or a cusp; viz., regarding a, b, c as current areal coordinates, we have here a 
conic - + t^ + - = 0, the locus of the centres of the variable circle, and the solution 
a b c 
depends on establishing a relation between this conic and the orthotomic circle or Jacobian 
of the three given circles. I have in my paper “ Investigations in connection with 
Casey’s Equation,” Quart. Math. Jour. vol. vm. (1867), pp. 334—342, [395] given, after 
Professor Cremona, a solution of the general question to find the number of the curves 
\/lU+ 4 JmV+\/nW=0 t which have a cusp, or which have two nodes, and I will here 
reproduce the leading points of the investigation. I remark, that although one of the 
loci involved in it is the same as that occurring in the case of the three circles 
(viz., we have in each case the Jacobian of the given curves), the other two loci 
2 and A, which present themselves, seem to have no relation to the conic of centres 
which is made use of in the particular case.