138 
PROFESSOR CAYLEY ON THE CURVES 
Annex No. 6 (referred to, No. 48). — Containing, with the variation referred to in the 
text , Zeuthen’s forms for the characteristics of the conics which satisfy four conditions. 
:: )= n -f- 2m, 
( .*./)= 2n -\-4rn, 
( ://)=4w + 4m, 
( ■ ///)=4w+2m, 
( ////)= 2n + 
^’ ^(.\) —2 m( m+ 3) + r, 
(: /) = 2m( m+2%— 5) + 2r, 
( • //) =2w(2m+ 5)4-2<S, 
(///) —2n ( m+ 3)+ &; 
^(:) =-|-[2m 3 -(-6m 2 w— w 3 — 30m 2 — 18m%-)-13w 2 -l-84m— A2n-\-(fm-\-?>n— 26)r], 
( • /) =^[(m-[-w)(— (m+w) 2 — 7(m-fw) + 48) -}-4mw(3m-|-3w— 13)-|-2(3 to- 1-3#— 20)(&+r)], 
(//) =^[— m 3 + 6mw 2 -l-2M 3 -l-13m 2 — \8mn— 30n 2 — 42m-j-84w+(3m-l-6w— 26)S] ; 
(1? 1? 1> 1)^ . ) = -i{ 2(m— 3)(m— 4)(% 2 — m— n)-\-(n— 3){n— 4)(m 2 — m— n) 
+ 4( m 2_ll m+ 28)r +2(^ 2 — 11^H-28)S 
+ (4(ra - 4){m - 4) - l)(2i + r) + 2S 2 + r 2 } , 
(/) =i{ (m—3)(m—4)(n 2 —m—n) +2(n—3)(n—4)(m 2 —m—n) 
+ 2(m 2 -llm+28)r +4(w 2 -llw+28)S 
+ (4 (n - 4)(m - 4) - l)(i + 2r) + S 2 + 2r 2 } ; 
(2) (-'.)= ^ + ;, 
(:-/)=2(3w+i), 
( ' //)— 2(3m+/), 
(///)= 3m+ <; 
^ (:)=3(2mw-J-w 2 +4m— 10w)-{-(2m-[-w— 14)«, 
( • /) —2(3m-4-i)(m-\-n— 12) +24(m-j-w), 
(//) = 3(m 2 +2mw— 10m-\-4n)-\-(m-i~2n-- 14) i 
(2, 1, 1)( . ) = (2m+^-7)(6r+(w-3» 
+ {{m — n)[m + n — 5) + r){3m + / — 36) 
+ 12(m— n){m-\-n— 3), 
(/) = ( m+ 2w-7)(6H(m-3» 
+ ((n — m)(m -f n — 5) + &)(3m -f / — 36) 
+ 12{n— m)(m+n— 3); 
