ME.  A.  CAYLET’S  FIETH  MEMOIE  UPON  QUANTICS. 
451 
conditions  which  imply  that  the  several  determinants 
^{ac—¥),  2>ad — he,  ae-\-2bd — 3c^  2>{be—cd),  6(ce — , 
a , h , c , d , e 
all  of  them  vanish.  If  for  a moment  we  write  6H=(«',  b',  c',  d',  y)*,  then  the  deter- 
minants are 
I b',  c',  d! , e'  , 
I a,  b,  c,  d,  e 
we  have  identically 
ad' —a!d=?>{bc'—b'c), 
eb'  -e'b  = ^dc'-d'c), 
ae'  — a!  e =S(bd'—b'd), 
and  the  ten  determinants  thus  reduce  themselves  to  seven  determinants  only,  these  in 
fact  being,  to  mere  numerical  factors  the  coetheients  of  the  cubicovariant ; this 
perfectly  agrees  with  a subsequent  result,  viz.  that  the  cubicovariant  vanishes  identi- 
caDy  when  the  quartic  is  a perfect  square. 
137.  It  may  be  remarked  that  the  equation  6IH  — 9 JU  = 0 will  be  satisfied  identically  if 
a= -^1  e—-^ibd—{c — (p)(c+2<p), 
c — <p  c — fS  ' 
where  (p  is  arbitrary ; the  quartic  is  in  this  case  the  square  of 
If  with  the  conditions  in  question  we  combine  the  equation  1 = 0 (which  in  this  case 
implies  also  J = 0),  we  obtain  p=0,  and  consequently 
abed 
b c d e’ 
or  the  quartic  will  be  a complete  fourth  power. 
It  is  easy  to  express  in  terms  of  the  coefficients  a',  b',  c',  d',  e'  of  6H  the  difierent 
determinants 
a,  b,  c,  d !, 
b,  c,  d,  e 
we  have  in  fact 
ae—bd 
V(^'e'+4.b'd'-Se'^) , 
% 
1 
CO 
5 ac—b^ 
=ia', 
ad — be 
be—ed 
— ly 
3^  5 
^ee—d^ 
— Xp' 
— 6^  5 
