424 
JklE.  A.  CAYLEY’S  EOLITH  MEMOIR  IPOJs  QrAXTICS. 
or  what  is  the  same  thing,  the  value  of  the  discriminant  □ &c.)  is 
( — /3,  ...). 
It  would  have  been  allowable  to  define  the  discriminant  so  as  that  the  leading  tenn 
should  be 
in  which  case  the  discriminant  would  have  constantly  the  same  sign  as  the  product  of 
the  squared  differences ; but  I have  upon  the  whole  thought  it  better  to  make  the  lead- 
ing term  of  the  discriminant  always  positive. 
83,  A quantic  of  an  even  order  2^  has  an  invariant  of  peculiar  simplicity,  \iz.  the 
determinant  the  terms  of  which  are  the  coefiicients  of  the  ^th  differential  coefficients, 
or  derived  functions  of  the  quantic  with  respect  to  the  facients ; such  invariant  may  also 
be  considered  as  a tantipartite  invariant  of  the  ^th  emanants.  Thus  the  sextic 
(a,  J,  c,  d,  e,  /,  yf 
has  for  one  of  its  invariants,  the  determinant 
a, 
b. 
c, 
d 
b, 
d, 
e 
c. 
d, 
e, 
f 
d, 
e, 
f, 
9 
The  invariant  in  question  is  termed  by  Professor  Sylvestee  the  Catalecticant. 
84.  Professor  Sylvestee  also  remarked,  that  we  may  from  the  catalecticant  form  a 
function  containing  an  indeterminate  quantity  X,  such  that  the  coefficients  of  the  dif- 
ferent powers  of  X are  invariants  of  the  quantic ; thus  for  the  sextic,  the  function  in 
question  is 
a , h , c , d — X 
h ^ c , <^-j--g-X,  6 
c , d—^-k,  e , / 
6 , / ,9 
where  the  law  of  formation  is  manifest ; the  terms  in  the  sinister  diagonal  are  modified 
by  annexing  to  their  numerical  submultiples  of  X with  the  signs  + and  — alternately, 
and  in  which  the  multipliers  are  the  reciprocals  of  the  binomial  coefficients.  The 
function  so  obtained  is  termed  the  Lamhdaic. 
85.  If  we  consider  a quantic  of  an  odd  order,  and  form  the  catalecticant  of  the  penul- 
timate emanant,  we  have  the  covariant  termed  the  Canonisant.  Thus  in  the  case  of  the 
quintic 
{a,  b,  c,  d,  e,  /J^’,  yf, 
aw-\-by,  bw-j-cy,  cw-\-dy 
bx-\-cy,  €£c-\-dy,  dx-\-ey 
cx-\-dy,  dx-\-ey , ex  -\-fy 
the  canonisant  is 
