440 
ME.  A.  CAYLEY’S  EIFTH  ^^IEMOIE  L^POX  QrAXTICS. 
If,  for  example,  v=(3,  then  we  have 
And  again,  if  u=ci,  v=-a',  w=a!',  then  we  have 
Or  putting  T=0,  the  first  equation  gives 
and  the  second  equation  gives 
which  are  both  of  them  well-known  forms. 
114.  A corresponding  transformation  applies  to  the  equation 
1, 
1, 
1, 
aa 
= 0, 
/3,  /3',  (3f3' 
y,  y,  77' 
1, 
which  expresses  the  homography  of  two  pairs.  In  fact,  calling  the  determhiant  and 
representing  by  V the  similar  determinant 
ss  , 
— S, 
— S, 
1 
tf  , 
—t. 
1 
uu'. 
— u'. 
—u. 
1 
vv' , 
—v'. 
1 
which,  equated  to  zero,  would  express  the  homography  of  the  sets  (s,  t,  it,  r)  and 
(s',  t',  id,  v'),  we  have 
V^=  {s-a)(s'-a'),  (s-(i)(s'-(5'),  (s -y){s' -y'),  (s-^)(s'-y). 
(^t-a)(t'-cc'),  (t-(5)(t'-l5'),  (t-y)(t'-y),  (t-^)(d-h'), 
(u—a)(u' —a'),  (u~l3)(u'  — (3'),  (u—y)(u'  — y'),  (u—^)(u'—^')  j 
(v—«)(v' —a'),  (v—(5){v'  —(5'),  (v—y){v'—y'),  (v —h' ) \ 
which  gives  various  forms  of  the  equation  of  homography.  In  particular,  if  s = a,  s'=^'. 
t = f5,  d — cd,  u—y,  vl—'S,  v=b,  v'=.y,  then 
V^= 
(l3  — y)(cc'  — y'),  ((3  — ^)(oc'—^') 
«')  (y-/3)(^' -/3') 
(^-/3)(7'-/3') 
and  the  right-hand  side  breaks  up  into  factors,  which  are  equal  to  each  other  (whence 
also  V='^),  and  the  equation  "^=0  takes  the  form 
