PROFESSOR CAYLEY ON THE CURVES 
But I make another alteration in the form of his results ; he gives, for instance, the 
characteristics of (1, 1)(1)(1) as 
(Jj = + m 2 n^) -\-^n x n 2 , 
v —v" 1 m{ni 2 v" {m x n 2 + m 2 n l ) + v' n x n 2 , 
where 
l«/ =2 m( m+ n— 3)+ r, =(1,1.*.), 
^ =v' =2n( m-\-2n — 5)+2r, =(1,1 :/ ), 
(jj'"=v" =2n (2m+ n- 5)+2&, =(1,1.//), 
/ , =2»( m+ rc-3)+ &, =(1,1///), 
viz. the four components have really the significations (1, 1 .\) set opposite to them 
respectively ; and accordingly, instead of giving the formulse for the two characteristics 
of (1, 1)(1)(1), I give those for the four characteristics (1, 1 .*.), &c. of (1, 1), thus in 
every case obtaining formulse which relate to a single curve only. Subject to the last- 
mentioned variation of form, I give Zeuthen’s original expressions in Annex 6 ; hut 
here in the text I express them as above in terms of (m, n, a), viz. 
49. We have the formulse 
W( :: ) = w+2w, 
( .*./)=2%+4m, 
( : //)=4w+4m, 
( • ///)=4w+2m, 
( ////)=2n+2m; 
.\)=2m 2 +2wm+-^ 2 — 2m— \n— -fa, 
( :/)=2m 2 +4wm+ n 2 —2m— n— 3a, 
( • //)= m 2 -\-imn-\-2n 2 — to— 2«— 3«, 
( ///)=%m 2 -{-2mn-\-2n 2 — \m— 2n— -fa. 
(1,1,1) 
( : )=fm 3 +2m 2 w+ mw 2 +|» 3 - 2m 2 — Zmn— \n 2 — a( — 3m— fw+13), 
( •/ )—\m 2 -\-2m 2 n-\-2mn 2 -\-\n 2 — m 2 —kmn— n 2 — — 3to— 3^+20), 
( // )=im 3 4- m 2 n-\-2mn 2 -\-^n 2 — \m 2 — 3mn— 2n 2 — §to— 3%-f- 13); 
(1, 1,1,1) 
( • ) = ^to 4 + fmVi + toW -f- \mn 2 + - 2 \m 4 
— -fro 3 — 3to 2 w — 2tow 2 — — -^m 2 —21 raw- -^% 2 + 
+ a( — §to 2 — 3wm— -^-f-^+a.f, 
( / )=^ 4 -m 4 +|m 3 w+OTV-l-|mw 3 -f y^-w 4 
— ^m 3 — 2 to 2 w — 3 mn 2 — — ^-m 2 — 2 1 inn — ^-f 1 ^ 2 -f- -f- 
+a( — fro 2 — 3wm— ^n 2 ^-=p)-}-a 2 .§ ; 
