ME.  A.  CAYLEY’S  FIFTH  MEMOIE  UPON  QUANTICS. 
437 
y,  y';  S,  &c.  will  be  in  involution  if  each  of  the  determinants  formed  with  any  three 
lines  of  the  matrix 
1,  a-f-a',  aa! , 
1,  3+^',  /3i3', 
1,  r+r'?  ry'. 
1,  §+§',  ll’, 
&c. 
vanishes-:  but  this  being  so,  the  determinant 
which  is  equal  to 
1, 
05,  cc', 
acc' 
1, 
(3,  13', 
(3(3' 
1, 
7->  y'? 
yy' 
1, 
a. 
1, 
a-\-(x! , 
acc' 
i3, 
1, 
(3+(3', 
(3(3' 
y. 
1, 
y _|_y'^ 
77 
1, 
will  vanish,  or  the  two  sets  (a,  |3,  y,  S)  and  («',  (3',  y',  ^')  will  be  homographic  ; that  is,  if 
any  number  of  pairs  are  in  involution,  then,  considering  four  pairs  and  selecting  in  any 
manner  a term  out  of  each  pah’,  these  four’  terms  and  the  other  terms  of  the  same  four 
pairs  form  respectively  two  sets,  the  two  sets  so  obtained  wall  be  homographic. 
110.  In  particular,  if  we  have  only  three  pairs  a,  cc';  (3,  j3';  y,  y',  then  the  sets  a,  (3,  y,  a' 
and  a',  (3\  y',  a Avill  be  homographic ; in  fact,  the  condition  of  homography  is 
which  may  be  written 
or  what  is  the  same  thing. 
1, 
«, 
a'. 
cccc' 
1, 
i3, 
/3', 
(3(3' 
1, 
y. 
7\ 
yy' 
1, 
oo'. 
05, 
acc' 
05, 
1, 
05  + 
a', 
1, 
(3-\-(3 , 
y. 
1, 
y+y', 
a'. 
1, 
CC+'i 
a', 
= 0, 
a , 1,  a-\-cx!,  aa'  =0, 
P ? 1? 
7 5 y+y',  yy' 
a'  — a,  0,  0 , 0 
