432 
ME.  A.  CAYLEY’S  FIFTH  BFEMOIE  LPOX  QUAXTICS. 
and  also,  taken  negatively,  in  the  form 
4(a5'  - dV){l)d  - h'c)  - {ad  - a!c)\ 
which  is  the  discriminant  of  the  Jacobian;  and  in  the  form 
4(«c — Jf){a!d — — {ad —2hh'-\-  ca'f, 
Avhich  is  the  discriminant  of  the  Intermediate. 
98.  We  have  the  following  relations: — 
{a,  b,  c\Vx~\-dy,  —o!x—yyf= 
— {dd—V'^)  {a^h  ^ c 'Jx,  yf 
-\-{a  d —2hb' -\-cd)  («',  b\  d^x,  yf, 
{d,  h\  dyhx-\-cy,  —ax~hyf= 
-\-{ad —2hh’ -\-cd)  {a,  b,  dX^,  yf 
— {ac—d)  {d,  V,  dXx,  y)\ 
and,  moreover, 
{ac—d,  ad —Ibb' -\-cd ^ aV— U/ 
= — {{ab’—db,  ad—dc,  bd  — b'djx-,  yfy\ 
an  equation,  the  interpretation  of  which  will  be  considered  in  the  sequel. 
99.  The  most  important  relations  which  may  exist  between  the  two  quadrics  are — 
First,  when  the  connective  vanishes,  or 
ad —2bb'  -^-cd  =■(), 
in  which  case  the  tAvo  quadrics  are  said  to  be  harmonically  related:  the  nature  of  this 
relation  will  be  further  considered. 
Secondly,  when  11=0,  the  two  quadrics  have  in  this  case  a common  root,  AA’hich  is 
given  by  any  of  the  equations, 
x‘^\2xy\y'^='baldi  :B4ll  :BcR 
=B„,K  :B4,II 
= bd — Vc : cd  — da : ab' — db. 
The  last  set  of  values  express  that  the  Jacobian  is  a perfect  square,  and  that  the  two 
roots  are  each  equal  to  the  common  root  of  the  two  quadrics. 
The  preceding  values  of  the  ratios  x"^ : 2xy : y^  are  consistent  AAuth  each  other  in  virtue 
of  the  assumed  relation  11=0,  hence  in  general  the  functions 
4h„E.B,R-(^,R)^  &c. 
all  of  them  contain  the  Resultant  R as  a factor. 
It  is  easy  to  see  that  the  Jacobian  is  harmonically  related  to  each  of  the  quadrics ; in 
fact  we  have  identically 
a{bd— b'c)  -{-b{cd— da) + c {ab' —db)=0, 
d{bd — b'c)  -f-  b'{  cd —<?'«)+ d{  ab' — a'i) = 0, 
which  contain  the  theorem  in  question. 
