98 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



the corresponding trizomal is thus a conic, and it has been seen that the other 

 trizomal is a cubic. 



VII. If we have 1, 1, 1, 1 =0, and(l, 1, 1, l ) ( Ji,Jm,Jn, J P )=0 



", b, c, d 

 a 2 , b 2 , c 2 , d 2 



-'. b"\ r'"\ d" 2 



the tetrazomal has a branch ideally containing (z s = 0) the line infinity 3 times : 

 order is = 5 ; orders of the trizomals are 3, 2. We have here 



1, 



1, 1, 1 



a, 



b, c, tl 



a 2 , 



b 2 , c 2 , d 2 



a"*, 



b" 2 , c" 2 , d" 2 



and thence 



which give 

 Moreover 



V I '■ ym : \/ n : V ' p = a : b : c : d , 

 JT[ = a + d , JT a = a + d 



JS, = b - Jf , Jn, = b + Jf 



*/T x + y/m^ + \A7 1 = , \/l, + Vnh, + Vn. 2 = 

 aJT^ + b J^ + c J^ = a (a + d) + &b+ ec 



^ a 



= (a-d)d-(b-c) J b Z d 



^ a 



and similarly 



aJT 2 +bJ^ 2 +cJn 2 =d|(a-d) + (6-c) N /g } ; 



whence in virtue of 



ad _ (b - c) 2 

 be ~ (d-a) 2 ' 



one of the two expressions is = ; and the trizomals are thus a conic and a 

 cubic. 



The Decomposable Curve ; Transformation to a different set of Coney clic Foci — Art. No. 206. 

 206. Consider the decomposable case of 



JTA + JmB + JnC + <JpO = ; 



