ME.  A.  CAYLEY’S  EOIJETH  MEMOIE  UPON  QUAJ^fTICS.  417 
and  a covariant  of  XU +jW<V,  if  in  such  covariant  we  treat  X,  p as  facients,  will  be  a quantic 
of  the  form 
(A,  B,  ..  B\  A'lx,  i^r, 
where  the  coefficients  (A,  B,  ..  B\  A')  will  be  covariants  of  the  quantics  U,  V,  viz.  A 
will  be  a covariant  of  the  quantic  U alone ; A'  vrill  be  the  same  covariant  of  the  quantic 
V alone,  and  the  other  coefficients  (which  in  reference  to  A,  A'  may  be  termed  the 
Connectives)  will  be  covariants  of  the  two  quantics ; and  any  coefficient  may  be  obtained 
from  the  one  which  precedes  it  by  operating  on  such  preceding  coefficient  with  the 
combinantive  operator 
or  from  the  one  which  succeeds  it  by  operating  on  such  succeeding  coefficient' with  the 
combinantive  operator 
... 
the  result  being  divided  by  a numerical  coefficient  which  is  greater  by  unity  than  the 
index  of  fju  or  (as  the  case  may  be)  X in  the  term  corresponding  to  the  coefficient  operated 
upon.  It  may  be  added,  that  any  invariant  in  regard  to  the  facients  (X,  (jb)  of  the  quantic 
(A,  B,  ..B',  A'lX, 
is  not  only  a covariant,  but  it  is  also  a combinaiit  of  the  two  quantics  U,  V. 
As  an  example,  suppose  the  quantics  U,  V are  the  quadrics 
(«,  5,  cjx,  yf  and  («',  V,  y)\ 
then  the  quadrinvariant  of 
XU+|M-V  is 
which  is  equal  to 
{ac—Jf,  ad —2hV  cdd—V^^^,  (Jbf, 
and  ad —2lh’  -^ca!  is  the  connective  of  the  two  discriminants  ac—lf  and  dd—V^. 
65.  The  law  of  reciprocity  for  the  number  of  the  invariants  of  a binary  quantic*, 
leads  at  once  to  the  theorems  in  regard  to  the  number  of  the  quadrinvariants,  cubinva- 
riants  and  quartinvariants  of  a binary  quantic  of  a given  degree,  ffi’st  obtained  by  the 
method  in  the  second  part  of  my  original  memoir  f.  Thus  a quadric  has  only  a single 
invariant,  which  is  of  the  degree  2 ; hence,  by  the  law  of  reciprocity,  the  number  of 
quadrinvariants  of  a quantic  of  the  order  m is  equal  to  the  number  of  ways  in  which  m 
can  be  made  up  with  the  part  2,  which  is  of  course  unity  or  zero,  according  as  m is  even 
or  odd.  And  we  conclude  that 
The  quadrinvariant  exists  only  for  quantics  of  an  even  order,  and  for  each  such 
quantic  there  is  one,  and  only  one,  quadrinvariant. 
66.  Again,  a cubic  has  only  one  invariant,  which  is  of  the  degree  4,  and  the  number 
of  cubinvariants  of  a quantic  of  the  degree  m is  equal  to  the  number  of  ways  in  which 
m can  be  made  up  with  the  part  4.  Hence 
* Introd.  Memoir,  No.  20.  f Ibid.  Nos.  10-17. 
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