448 
ME.  A.  CAYLEY’S  ELFTH  MEMOIR  UPOX  QUAXTICS. 
We  have,  in  like  manner, 
P-Q, 
P-^y^Q, 
— OTgjCIH  — 'i3J'2  JU)  = iy^P  — <yQ, 
and 
— Pq  — Qo? 
Po— 
^(co— a^)  (ts'i  — 7^2)  = iyTo — tijQo, 
and  therefore 
(^,-^s)\/iH-^,JU=(.y-c,^)y-(P-Q)  (Po-Qo), 
(otj — IH — TiTg  JU = (a< — — (<yP — <y^Q)(iyPo — 
(7^3  - z7,)x/IH  - 7;r3  JU  = (o;  - - (^yT  - <yQ)(a;To  - o^Qo) ; 
and  hence  disregarding  the  common  factor  u — <y^,  the  square  of  the  linear  factor  of  the 
quartic  is 
\/  — (P  — Q)(Po— Qo)  + \/  — (iwP  — )(iyPo — <y^Qo)  + \/  — (cyT  — iyQ)((a»T„  — jyQt,) , 
which  is  the  required  solution. 
It  may  be  proper  to  add  that 
T^j—  Pq-|“  Qo? 
— rs^—u  Po  + iy‘^Qo, 
— 75'3=(yTo+<x;  Qo- 
132.  The  solution  gives  at  once  the  canonical  form  of  the  quartic;  in  fact,  wu-iting 
X + < Y = 2>/( ^2  — 7I5-3)(to-3  — TO-,)  JX^ 
X— <Y  = 2\/  (73-2  — 73-3)(7D-3  — 75-i)x/jy% 
where  X,  Y have  their  former  significations,  we  find,  by  a simple  reduction, 
IH-77.,JU=  (^3_^,)J(x^  + y^)^ 
IH-77r2JU=  -(t32-733)J(x^-y^)^ 
IH-733JU  = - K-^3)(^3-?i)  J _ 4^2 
and  thence  putting 
73-g  _■§■(«)  — a;^)(w^Po  + cyQo) 
TOj— TOg”  (cO^Pq  — CoQq)  ’ 
we  have 
U = x"* + y ^ -h  6 
which  is  the  form  required. 
133.  The  Hessian  may  be  written  under  the  form 
So,  — S4,  y)H, 
that  is,  as  the  evectant  of  the  cubinvariant. 
