ME.  A.  CAYLEY’S  FIFTH  MEMOIE  UPON  QUANTICS. 
455 
these  values  give  also 
6IH-9JU=0. 
146.  In  the  case  of  three  equal  roots,  we  have 
a"'U ={x—a,yY{x—ly% 
I =0,  J=0,  0=0, 
a-^'R=—^{a—lY{2{x—lyf-\-{x— ayf  ){x— ooyf, 
i^{u  — lf{x^(x,yf; 
and  in  the  case  of  four  equal  roots,  we  have 
a~^\]  =[x—ayy, 
I =0,  J=0,  0=0, 
H=0,  0=0. 
The  preceding  formulae,  for  the  case  of  equal  roots,  agree  with  the  results  obtained  in 
my  memoir  on  the  conditions  for  the  existence  of  given  systems  of  equalities  between 
the  roots  of  an  equation. 
Addition,  7tli  October,  1858. 
Covariant  and  other  Tables  (binary  quadrics  Nos.  25  bis,  29  a,  49  a,  and  50  bis). 
Mr.  Salmox  has  pointed  out  to  me,  that  in  the  Table  No.  25  of  the  simplest  octin va- 
riant of  a binary  quintic*,  the  coefficients  — 210,  —17,  +18  and  +38  are  erroneous, 
and  has  communicated  to  me  the  corrected  values,  which  I have  since  verified : the 
terms,  with  the  corrected  values  of  the  coefficients,  are — 
No.  25  bis. 
— 220  ah<?dref 
+ 22 
+ 74  bc^d^e 
- 27  acY' 
Mr.  Salmon  has  also  performed  the  laborious  calculation  of  Heemites’  18-thic  invariant 
of  a binary  quintic,  and  has  kindly  permitted  me  to  publish  the  result,  which  is  given 
in  the  following  Table : — 
No.  29  a. 
+ 1 a'dY 
+ 60  dbccP^Y 
— 10  dbd&Y 
+ 120  ddde*/* 
— 15  dcdY 
- 5 
-90  dbcd-eY 
-10  dbddf 
— 40  ddd’Y 
— 110  dcd^e^Y 
+ 10  d'd^d'f^ 
+ 60  dbcdeY 
+ 5 
+ 60 
+ 265  dcd*eY 
-10  ddr-Sf 
— 15  dbceY 
- 1 ddf 
+ 30  ddd^eY 
— 200  dcd^eY 
+ 5 u'deY 
+ 10  dbdeY 
+ 15  dddeY 
— 180 
+ 65  dcdPey 
- 1 ff'e’y 
-35  dbd'dY 
— 10  dddf 
+ 120  ddde^Y 
— 10  dcdd'* 
-15  a%cdY 
V ; 
+ 40  dbtPeY 
— 90  dd(PeY 
- 20  aVey 
J 
+ 45  dd?ef 
* Second  Memoir,  Philosophical  Transactions,  t.  cxlvi.  (1856)  p.  125. 
MDCCCLVIII.  3 P 
