444 
ME.  A.  CAYLEY’S  EIETH  MEMOIE  UPO^’'  QrAXTICS. 
1 
— (3y — ya  — 05j3, 
= — 6c(^y—(3y^—yu^  — af3^—f3y—y^ci  — a%  '(x,  yf 
+yW+«^(3^ — a^/3y — |3^ya  — 7^“/^  J 
= —¥{(«  + "/^ + +((^y  + "7“ + } { (a  + co^(^-^uy)x-\-{(By + ^y'ya -f  a;a^)j^ } 
(where  a is  an  imaginary  cube  root  of  unity), 
« - 3 $ = 2 («  - /3 ) («  - y ) 2 (a:  - /3y)  2 (a?  - yy) 
2(a^  + /3®  + y^)  — 3(/3y2  + ya2  + «/32  + /32y  + y2a-l- a2(3)  + 12a/3y,  " 
— 2 (a^/3y  + /S^ya  + y2«/3)  + 4 ((S^y^  _f.  _|_  ^2^2^  _ ^ ^ ^133  _j_  ^3^  _|_  _|.  ^3^)^ 
j — 2 (a/32y2  -f  (3y2«2  4-  ^ 4 («®/3y  + /3^y«  + y®a/3)  — (/S^y^  + y2«3  4.  ^2^  _|_  ^3^2  _|_  ^^,^2  4.  *3^2 
L + 2 (/3^y®  + y®a^  + a®j6®)  — 3 (a/S^y^  4-  /Sy^a^  4-  ya2|33  4.  ^^3^2  4.  |3,y3^2  _j_  4- 1 2a.^^-y^ 
= {(2a  — /3  — y)a;  + (2/3y  — y«  — a/3)y}  { (2/3— y — a)a7+  (2ya— a/3— /3y)y}  {(2y  — a — |3)a;  + (2a/3  — py 
123.  It  may  be  observed  that  we  have  «~®nU^=  — which,  with  the  above 
values  of  H,  O in  terms  of  A,  B,  C and  the  equation  A+B-f-C=0,  verifies  the  equation 
O^—  □U^+4H®=0,  which  connects  the  covariants.  In  fact,  we  have  identically, 
(B-C)2(C-A)^(A-B)^= 
-4(A+B+C)^ABC+(A+B+C)^(BC+CA+AB)^+18(A+B+C)(BC+CA+AB).ABC 
-4(BC+CA-fAB)^-27A^B^e, 
by  means  of  which  the  verification  can  be  at  once  effected. 
124.  If,  as  before,  a is  an  imaginary  cube  root  of  unity,  then  we  may  write 
27a-^0  = -(B-  C)(C- A)(A-B) 
27«-^Ux/'n=  3(a— a;")ABC, 
and  these  values  give 
27q/  ^■^(^“1~U'\X d ) — { (® “l“<v^i3-l~*^7)  ^~l~(/3y-|-ai^ya-|-aia/3  )^}^ 
27<2  ®^(^ — Uv^ □ ) = {(afi-ift)/!  -\-u^y)x-\-((5y-^a)ya  -\-ct)^ci(5)yy, 
and  we  thence  obtain 
4/i(<i>+u  Va)—yi(^-v 
which  agrees  with  a former  result. 
125.  The  preceding  formulae  show  without  difficulty,  that  each  factor  of  the  cubi- 
covariant  is  the  harmonic  of  a factor  of  the  cubic  with  respect  to  the  other  two  factors 
of  the  cubic ; and  moreover,  that  the  factors  of  the  cubic  and  the  cubicovariant  form 
together  an  involution  haring  for  sibiconjugates  the  factors  of  the  Hessian.  In  fact,  the 
harmonic  of  cc—ay  with  respect  to  {x—^y){x—yy)  is  (2«— j3  — y)a’+(2|87— ya  — a|S)y,  j 
which  is  a factor  of  the  cubicovariant ; the  product  of  the  pair  of  harmonic  factors  is  ' 
