ME.  A.  CAYLEY’S  FIFTH  MBMOIE  UPON  QUANTICS. 
435 
the  pairs  (a,  (3),  {a',  j3')  as  given,  there  are  of  course  two  values  of  ^ which  satisfy  the 
preceding  equation ; and  calling  these  and  , then  and  are  said  to  be  the  sibi- 
conjugates  of  the  involution  a,  (3  ; a!,  j3'.  It  is  easy  to  see  that  are  the  roots  of  the 
equation  H=0,  where  H is  the  Jacobian  of  the  two  quadrics  U and  U'  whose  roots  are 
(a,  j3),  {a!,  /3').  In  fact,  the  quadric  whose  roots  are  6^^  is 
y\  2yx  , x' 
I , a 4-/3 , a/3 
I , a'+/3',  a'jS' 
which  has  been  shown  to  be  the  Jacobian  in  question.  But  this  may  be  made  clearer 
as  follows : — If  we  imagine  that  X,  are  determined  in  such  manner  that  the  inter- 
mediate XU  4- may  be  a perfect  square,  then  we  shall  have  XU-|-|aU'  = a"(A’— 
where  & denotes  one  or  other  of  the  sibiconjugates  6^,  6^^  of  the  involution.  But  the 
condition  in  order  that  XU  4"i“'U^  may  be  a square  is 
{ac—lf,  ad —2hh'  -\-ca! , J'^'^X,  yjf; 
and  observing  that  the  equation  X:|a  = U':  — U implies  XU4-(W-U'=0  = a"(a’— it  is 
obnous  that  the  function 
{ac—¥,  ad —2hV -\-cd ^ — U)^ 
must  be  to  a factor  equal  to  {x—^iyy'{x—6^yY.  But  we  have  identically 
{ac—d,  ad —2hV -\-ca! , —Vf=  — {{ab'  — o!b,  ad—a'c,  hd—h'd^x, 
and  we  thus  see  that  {x—6^y),  {x—d,^y)  are  the  factors  of  the  Jacobian. 
106.  It  has  been  already  remarked  that  the  Jacobian  is  harmonically  related  to  each 
of  the  quadrics  U,  U';  hence  we  see  that  the  sibiconjugates  0^,  of  the  involution  a,  /3, 
a!,  j3'  are  a pair  harmonically  related  to  the  paii-  a,  j3,  and  also  harmonically  related  to 
the  pair  a',  /3',  and  this  properly  might  be  taken  as  the  definition  for  the  sibiconjugates 
6^1  of  an  involution  of  four  terms.  And  moreover,  a,  (3 ; a',  (3'  being  given,  and  0^,  0^^ 
being  deteimined  as  the  sibiconjugates  of  the  involution,  if  a",  (3"  be  a pair  harmoni- 
cally related  to  0^ , , then  the  three  pairs  a,  |3 ; a',  f3' ; a",  (3"  will  form  an  involution ; 
or  what  is  the  same  thing,  any  three  pairs  a,  (3 ; a',  (3' ; a",  /3",  each  of  them  harmoni- 
cally related  to  a pair  0^,  0^,  will  be  an  involution,  and  0^,  0^^  will  be  the  sibiconjugates 
of  the  involution. 
107.  In  particular,  if  a,  (3  be  harmonically  related  to  0^,  0^,  then  it  is  easy  to  see  that 
0,,  0^  may  be  considered  as  harmonically  related  to  0^,  0^,  and  in  like  manner  0^^,  0^1  will 
be  hamonically  related  to  0^,  0^^-,  that  is,  the  pairs  0^\  ^^p  0^^  and  a,  f3  will  form  an 
involution.  This  comes  to  saying  that  the  equation 
1, 
1, 
1, 
20,  , 0^ 
= 0 
a-j-f3,  ccj3 
