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XXII,  A Fourth  Memoir  upon  Quantics.  By  Aethuk  Cayley,  Esq.^  F.B.S. 
Eeceived  February  11, — Eead  March  18,  1858. 
The  object  of  the  present  memoir  is  the  further  development  of  the  theory  of  binary 
quantics ; it  should  therefore  have  preceded  so  much  of  my  third  memoir,  t.  147  (1857), 
p.  627,  as  relates  to  ternary  quadrics  and  cubics.  The  paragraphs  are  numbered  con- 
tinuously with  those  of  the  former  memoirs.  The  first  three  paragraphs.  Nos.  62  to  64, 
relate  to  quantics  of  the  general  form  y,  . .)™,  and  they  are  intended  to  complete 
the  series  of  definitions  and  explanations  given  in  Nos.  54  to  61  of  my  third  memoir ; 
Nos.  68  to  71,  although  introduced  in  reference  to  binary  quantics,  relate  or  may  be 
considered  as  relating  to  quantics  of  the  like  general  form.  But  with  these  exceptions 
the  memoir  relates  to  binary  quantics  of  any  order  whatever;  viz.  No.  65  to  80  relate 
to  the  covariants  and  invariants  of  the  degrees  2,  3 and  4 ; Nos.  81  and  82  (which  are 
introduced  somewhat  parenthetically)  contain  the  explanation  of  a process  for  the 
calculation  of  the  invariant  called  the  Discriminant ; Nos.  83  to  85  contain  the  definitions 
of  the  Catalecticant,  the  Lambdaic  and  the  Canonisant,  which  are  functions  occurring  in 
Professor  Sylvester’s  theory  of  the  reduction  of  a binary  quantic  to  its  canonical  form  ; 
and  Nos.  86  to  91  contain  the  definitions  of  certain  covariants  or  other  derivatives  con- 
nected with  Bezout’s  abbreviated  method  of  elimination,  due  for  the  most  part  to  Pro- 
fessor Sylvester,  and  which  are  called  Bezoutiants,  Cobezoutiants,  &c.  I have  not  in 
the  present  memoir  in  any  wise  considered  the  theories  to  which  the  catalecticant,  &c. 
and  the  last-mentioned  other  covariants  and  derivatives  relate ; the  design  is  to  point 
out  and  precisely  define  the  difierent  covariants  or  other  derivatives  which  have  hitherto 
presented  themselves  in  theories  relating  to  binary  quantics,  and  so  to  complete,  as  far 
as  may  be,  the  explanation  of  the  terminology  of  this  part  of  the  subject. 
62.  If  we  consider  a quantic 
(«,  b,  ..Xx,  y,  ...)'" 
and  an  adjoint  linear  form,  the  operative  quantic 
(a,  h,  . . .f, 
or  more  generally  the  operative  quantic  obtained  by  replacing  in  any  covariant  of  the 
given  quantic  the  facients  {x^  y,  .,)  by  the  symbols  of  difierentiation  (B|,  b,,,  ...)  (which 
operative  quantic  is,  so  to  speak,  a contravariant  operator),  may  be  termed  the  Pro- 
vector ; and  the  Provector  operating  upon  any  contravariant  gives  rise  to  a contravariant, 
which  may  of  com’se  be  an  invariant.  Any  such  contravariant,  or  rather  such  con- 
travaiiant  considered  as  so  generated,  may  be  termed  a Provectant ; and  in  like  manner 
the  operative  quantic  obtained  by  replacing  in  any  contravariant  of  the  given  quantic 
MDCC'CLVIII.  3 K 
