450 
ME.  A.  CAYLEY’S  ELFTH  ^klEMOIE  UPOY  QrAYTICS. 
equation  («,  5,  c,  d,  e\x,  ^)^=0, 
we  have  identically  (X,  Y being  arbitrary), 
3 
{a,  b,  c,  d,  eXX,  y) 
Xy-Yx 
=n/-(B.,  -S.,  B.,  bJX,  Y)^A, 
+\/— (b„  — B,,,  — Bi,  B„XX, 
+n/-(B.,  -B„  B„  -B„  B.XX,  Y)^A3, 
a theorem  due  to  Aeojvhold.  I have  quoted  this  theorem  in  its  original  form  as  an 
application  of  the  lambdaic,  but  I remark  that 
-(B„  -B„  B„  -B„  BJX,Y)^A=-x(«,  ...JX,  Y)^-(«c-5^  ...XX,  Y)^=-?X’'-H' 
if  U',  H'  are  what  U,  H become,  substituting  for  {x,  y)  the  new  facients  (X,  A').  Blore- 
over,  we  have 
for  substituting  this  value  in  the  equation  A = 0,  we  obtain  the  before-mentioned  equa- 
tion — M(ti7 — 1)  = 0.  We  have,  therefore, 
-(3„  -3„  -3„  S.IX,  Y)‘A=^U'-H'=-l(lH'-J=rU'), 
and  the  equation  becomes 
3 
Moreover,  if  {pc— ay)  be  a factor  of  the  quartic,  then  replacing  in  the  formula  y by  the 
value  a^,  {x,  y)  will  disappear  altogether ; and  then  changing  (X,  IT)  into  (x,  y)  where 
X,  y are  now  arbitrary,  we  have 
3 
{a,  b,c,  d,ejx,  yja,  l)yi:Y=x/lH-ziT,JU+-v/lH-;g,JU+x/lH-^3JU, 
X uy 
which  is  a form  connected  with  the  results  in  Nos.  130  and  131. 
136.  We  have 
— ^:xy‘^^ 
— 4^®^, 
x^ 
a , 
35  , 
3c  , 
d 
a , 
U , 
3c  , 
d , 
h , 
3c  , 
U , 
e 
A 
3c  , 
od  , 
^ , 
it  will  appear  from  the  formulse  relating  to  the  roots  of  the  quartic,  that  the  expression 
6IH  — 9JU  vanishes  identically  when  there  are  two  pair’s  of  equal  roots,  or  what  is  the 
same  thing,  when  the  quartic  is  a perfect  square.  The  conditions  in  order  that  the 
expression  may  vanish  are  obviously 
6{ac—b^):  S{ad—bc):  ae-\-2bd — 3(?^;  ^be — cd):  6(ce—d^):  9J 
==  a : b : c : d : e : 1, 
