96 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



and then 



{d — a) J I — (b — d) Jm + (c — d) Jn , 

 (d — a) Jp = (a — b) Jm + {a— c) Jn , 



whence 



b — c 

 d Jl- aj p - j^ a (b J n - Jm) , 



and we have thus 

 And similarly 



Jl, + 75, + Jn, = jj^ + Jg) (bj-,-, ^ . 



(observe that in the case not under consideration bjn — cjm = 0, and therefore 

 Jl x + Jni x + Jn x = 0, JT 2 + Jm., + Jn 2 = 0). In the present case we have 



a:b:c:d = (b-c)(c-d)(d-b):-(c-d)(d-a)(a-b):-(d-a)(a-b)(b-d):-(a-b)(b-c)(c-a), 



and thence 



ad (6 - c) 2 



bc~ (d - a) 2 ' 



so that only one of the two sums Jl x + Jm x + Jn v J I., + Jm., + Jn., is = 0, 



viz., assuming 



/ad h — c 

 •v be ~ d~^a ' 



we have 



Jl x + v^i "+" J^i = o . 

 And then also 



a V^i + * Vm^ + c */«! = a n// + h Jm + c Jn 



but we find 



and thence 



5 _ c _ 



dd J J - aa Jp = — - (6b Jn - cc J m ) , 



aj\ + bjm i +0^=^1-1 _ JH) (6bV» - ocVm) ,= , 



in virtue of J~ = , _ ' . Hence J\ : *//»! : x/w r = b — c : c — a: a — b, or the 

 corresponding trizomal is a conic, but the other trizomal is a quartic. 



