420 
ME.  A.  CAYLEY’S  EOTJETH  3kIEMOIE  LPOX  QTJAXTICS. 
72.  It  may  be  observed  also,  that  in  the  case  where  a tantipartite  invariant,  when  the 
several  quantics  are  put  equal  to  each  other,  does  not  become  equal  to  zero,  we  may 
pass  back  from  the  invariant  of  the  single  quantic  to  the  tantipartite  invariant  of  the 
system ; thus  the  lineo-linear  invariant  ad — 1W-\-cd  of  two  quadrics  leads  to  the  quadrin- 
variant  ac — Tf  of  a single  quantic;  and  conversely,  from  the  quadrin variant  ac — of  a 
single  quadric,  we  obtain  by  an  obvious  process  of  derivation  the  expression  ad —2hh' -\-c(d 
of  the  lineo-linear  invariant  of  two  quadrics.  This  is  in  fact  included  in  the  more  general 
theory  explained.  No.  64. 
73.  Eeverting  now  to  binary  quantics,  two  quantics  of  the  same  order,  even  or  odd, 
have  a lineo-hnear  invariant.  Thus  the  two  quadrics 
{a,  h,  cX^,  yf,  {(d,  b',  djx,  yf 
have  (it  has  been  seen)  the  lineo-linear  invariant 
ad —^hV  -\-ca'; 
and  in  like  manner  the  two  cubics 
{a,  h,  c,  djx,  yf,  (a',  V,  d,  d’Xx,  yf 
have  the  lineo-linear  invariant 
ad’ — ?>hd  -f-  — d(d, 
which  examples  are  sufficient  to  show  the  law. 
7 4.  The  lineo-linear  invariant  of  two  quantics  of  the  same  odd  order  is  a combinant, 
but  this  is  not  the  case  with  the  lineo-linear  invariant  of  two  quantics  of  the  same  even 
order.  Thus  the  last-mentioned  invariant  is  reduced  to  zero  by  each  of  the  operations 
and 
but  the  invariant 
is  by  the  operations 
and 
a' + 6' c' 4*  ^ d' 
ad  —2hh'-\-co! 
reduced  respectively  to 
2{ac-~b‘^) 
and 
2(a'd-b'^). 
75.  For  two  quantics  of  the  same  odd  order,  when  the  quantics  are  put  equal  to  each 
other,  the  lineo-linear  invariant  vanishes ; but  for  two  quantics  of  the  same  even  order, 
when  these  are  put  equal  to  each  other,  we  obtain  the  quadrinvariant  of  the  single 
quantic.  Thus  the  quadrinvariant  of  the  quadric  (a,  h,  dfjx,  yf  is 
ac—h^; 
and  in  like  manner  the  quadrinvariant  of  the  quartic  {a,  h,  c,  d,  ejjc,  y)*  is 
ae—4:bd-\-Sc’‘. 
