108 

 which is 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



= 2(a - ft) (ft - c) (c - a) [(ft - r)a + (c - «)jS + (ft - b)y] , 



and for the last term, omitting the factor 20, this may be deduced therefrom by 

 writing (a 2 , & 2 , c 2 ) in place of («, /3, 7), viz., it is 



= - 2(a- ft) 2 (ft - c) 2 (c - ft) 2 • 



Hence, restoring the omitted factors, and collecting, we find 



Norm {(b - c) J a 2 + 6 + a + (c - «) JW + t + p + (a - b) Jc 2 + 6 + 7} 



- (&- c )*«2+(c-a) 4 ^ z +(a-6)y-2(c-a) a (a-&) a /8y-2(a-5)" (ft- c )2y a _2(&- c )2 ( c -a) 2 a/3 

 + 4<«-ft) (&— c) (c-«) [(6-c)« + (c-cr)j8+(a-6)y] 

 + 4 (a— 6) (ft — c) (c — a) [(&c(6 — c)a + c«(c — fl)/3+fl6(a — 6)7] 

 - 4tf(a-6) s (6-c) 3 (c-a) 2 . 



Hence, first writing a — x, b — cc, c — a: in place of a, 6, c ; then # 2 for 6, and 

 (— a" 2 ,— 6" 2 ,— c" 2 ) for (a,/8, 7); and finally introducing z for homogeneity, we 

 find 



Norm {(ft - c) J(x - azf + y 2 - a"*z 2 + (c - a) J7+ (a - b) J~} = s 2 into 



g2((6 - c)V' 4 + (c - ft) 4 ft" 4 + (« - ft)V' 4 



- 2(e - a) 2 (ft - &) a &"V - 2 (« - ft) 2 (6 - c)W* - 2 (ft - c) 2 (c - afaV) 

 -4?/ 2 (ft - c) (c — ft) (ft - ft) [ (ft - c) ft" 2 + (c - a)b" + (ft — ft)c"2] 

 -4 (ft - f) (c -a) (a- ft) { (ft— c)ft" 2 («*&!- SB (6 + c) + aJ») 



+ (c — ft) ft" 2 (2 3 cft — zx (c + ft) + a; 2 ) 

 +(rt — ft) e" 2 (z*ab — zx(a + b)+ a; 2 )} 

 -4// (ft — cf (c -a) (a — 6) 2 . 



so that the equation (b — c) Jfip + (c — a) */W + (a — b) VC 5 = 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(ft - ft) 2 (ft - c) 2 (c - «) 2 (y - *V) , 



viz., when the three circles have each of them the same radius k", the curve is the 

 pair of parallel lines y 1 — k" 2 z 2 = ; and in particular when k " = , or the 

 circles reduce themselves each to a point, then the curve is y 2 — , the axis twice. 



Annex IV. — On the Trizomal Curves */lU+ *fmV + *JnW=- 0, which have a Cusp, or 



two Nodes. 



The trizomal curve JJlJ + ,J m V+ sfnW— 0, has not in general any nodes 

 or cusps: in the particular case where the zomal curves are circles, we have 



