416 
ME.  A.  CAYLEY’S  EOUETH  IklEMOIE  I7POY  QrA2m;CS. 
the  facients  (^,  f] by  the  symbols  of  differentiation  (B,,  ...)  (which  operative 
quantic  is  a covariant  operator),  is  termed  the  Contraprovector ; and  the  contraprovector 
operating  upon  any  covariant  gives  rise  to  a covariant,  which  may  of  course  be  an  inva- 
riant. Any  such  covariant,  or  rather  such  covariant  considered  as  so  generated,  may  be 
termed  a Contraprovectant. 
In  the  case  of  a binary  quantic, 
(«,  h,  ..Jcc,  yf, 
the  two  theorems  coalesce  together,  and  we  may  say  that  the  operative  quantic 
(«,  h,  — B^)*”, 
or  more  generally  the  operative  quantic  obtained  by  replacing  in  any  covariant  of  the 
given  quantic  the  facients  [x,  y)  by  the  symbols  of  differentiation  (B^,  — B,)  (which  is  in 
this  case  a covariant  operator),  may  be  termed  the  Provector.  Amd  the  Provector 
operating  on  any  covariant  gives  a covariant  (which  as  before  may  be  an  invariant),  and 
which  considered  as  so  generated  may  be  termed  the  Provectant. 
63.  But  there  is  another  allied  theory.  If  in  the  quantic  itself  or  in  any  covariant 
we  replace  the  facients  {x,  y,  ...)  by  the  first  derived  functions  (B^P,  B,P...)  of  any  con- 
travariant  P of  the  quantic,  we  have  a new  function  which  will  be  a contravariant  of 
the  quantic.  In  particular,  if  in  the  quantic  itself  we  replace  the  facients  {x,  y...)  by 
the  first  derived  functions  (B^P,  B^P,  . .)  of  the  Eeciprocant,  then  the  result  will  contain 
as  a factor  the  Eeciprocant,  and  the  other  factor  will  be  also  a contravariant.  And 
similarly,  if  in  any  contravariant  we  replace  the  facients  (^,  . .)  by  the  fii’st  derived 
functions  (B^W,  'd^W,  ..)  of  any  covariant  W (which  may  be  the  quantic  itself)  of  the 
quantic  U,  we  have  a new  function  which  will  be  a covariant  of  the  quantic.  Ajid  in 
particular  if  in  the  Eeciprocant  we  replace  the  facients (^,  ...)  by  the  fii*st  derived 
functions  (B^U,  B^,!!..)  of  the  quantic,  the  result  will  contain  U as  a factor,  and  the 
other  factor  will  be  also  a covariant.  In  the  case  of  a binary  quantic  (a,  b,  . yT 
the  two  theorems  coalesce  and  we  have  the  following  theorem,  riz.  if  in  the  quantic  U or 
any  co variant  the  facients  (x,  y)  are  replaced  by  the  first  derived  functions  (B^W,  — B,W) 
of  a covariant  W,  the  result  will  be  a covariant ; and  if  in  the  quantic  U the  facients 
[x,  y)  are  replaced  by  the  first  derived  functions  (B^,!!,  — B^.U)  of  the  quantic,  the  result 
will  contain  U as  a factor,  and  the  other  factor  mil  be  also  a covariant. 
Without  defining  more  precisely,  we  may  say  that  the  function  obtained  by  replacing 
as  above  the  facients  of  a covariant  or  contravariant  by  the  fii’st  derived  functions  of  a 
contravariant  or  covariant  is  a Trmismutmit  of  the  first-mentioned  covariant  or  contra-  ' 
variant. 
64.  Imagine  any  two  quantics  of  the  same  order,  for  instance,  the  two  quantics  ; 
u=(a,  ...X^,  y..)”*, 
Y={a',b’,  ...X^\y..y\  , 
then  any  quantic  such  as  XU.+iW'V  may  be  termed  an  Intermediate  of  the  two  quantics ; | 
