436  ME.  A.  CAYLEY’S  FIFTH  jHEMOIE  FTOX  QUA^sHICS. 
is  equivalent  to  the  harmonic  relation  of  the  pahs  a,  (3;  6^,  6^^',  and  in  fact  the  deter- 
minant is 
which  proves  the  theorem  in  question. 
108.  Before  proceeding  further,  it  is  proper  to  consider  the  equation 
1,  a,  ci',  aa!  =0, 
1,  i3,  ,S',  (3(3' 
1,  7,  r',  77' 
1,  h,  h',  hh’ 
which  expresses  that  the  sets  (a,  (3,  y,  h)  and  («',  (3',  §')  are  homographic ; for  although 
the  homographic  equation  may  be  considered  as  belonging  to  the  theory  of  the  bipartite 
qnadric  [x—ay)  (x— a'y),  yet  the  theory  of  involution  cannot  be  completely  discussed 
except  in  connexion  with  that  of  homography.  If  we  write 
K—{(3—y){ci—l),  B={y  —a){(3  C =(«  — /3  Xy  — S ), 
A'  = {(3'-y'){c.'-n  = C'  = {cc' 
then  we  have 
and  thence 
A+B+C=0, 
A'+B'-fC'=0, 
BC'-B'C=CA'-C'A=AB'~A'B  ; 
and  either  of  these  expressions  is  in  fact  equal  to  the  last-mentioned  determinant,  as 
may  be  easily  verified.  Hence,  when  the  determinant  vanishes,  we  have 
A:B:C=A:B':C'. 
Any  one  of  the  three  ratios  A : B : C,  for  instance  the  ratio  B : C= 
(y-«)(/3-S) 
(«— /3)  (y— S)’ 
is  said  to  be  the  anharmonic  ratio  of  the  set  {a,  (3,  y,  ^),  and  consequently  the  two  sets 
(a,  (3,  y,  and  (a',  (3',  y',  h')  will  be  homographically  related  when  the  anharmonic  ratios 
(that  is,  the  corresponding  anharmonic  ratios)  of  the  two  sets  are  equal. 
If  any  one  of  the  anharmonic  ratios  be  equal  to  unity,  then  the  four  terms  of  the  set 
g 
taken  in  a proper  manner  in  pairs,  will  be  harmonics ; thus  the  equation  q = 1 gives 
(y— «)(/3  — S) 
(a-/3)(y-8)  ’ 
which  is  reducible  to 
2ah-\-2(3'y — (a-l-§)(j3-|-y)=0, 
which  expresses  that  the  pairs  a,  § and  j3,  y are  harmonics. 
109.  Now  returning  to  the  theory  of  involution  (and  for  greater  convenience  taking 
a,  a'  &c.  instead  of  a,  (3  &c.  to  represent  the  terms  of  the  same  pair),  the  pairs  a,  a';  (3,(3'; 
