539 
156] A FIFTH MEMOIR UPON QUANTICS. 
If, for example, u = oc, v = (3, then we have 
T (« — /3) =— (a--oc') (oc - /3') (¡3-oc") (¡3-/3") + (/3 - «') (/3 - /3') («■-oc") (oc - /3"); 
and again, if u = a, v = a', w = a", then we have 
T = — (a - /3") (oc' - /3) (a" - /3') + (oc - /3') (a' - /8") (a" - /3). 
Putting T = 0, the two equations give respectively 
( a -«')(/3-«")_(a-/3")(/3-/3') 
(oc- a") («'-£) (a-^Xr -ZS)’ 
(a - /8") (a' - /8) (a" - /3') = (a - /3') (a' - /3") (a" - /8), 
which are both of them well-known forms. 
114. A corresponding transformation applies to the equation 
1, 
a, 
a', 
aoc' 
= 0, 
1, 
/3, 
/3', 
/3/3' 
1, 
7> 
/ 
7 > 
yy 
1, 
8, 
S', 
88' 
which expresses the homography of two pairs. In fact, calling the determinant \f r and 
representing by V the similar determinant 
ss', 
-s', 
-S, 
1 
tt' , 
-if, 
-t, 
1 
uu, 
— v!, 
— u, 
1 
vv', 
- V, 
- V, 
1 
which, equated to zero, would express the homography of the sets (s, t, u, v) and 
(s', t', u', v), we have 
VV = 
(s - a) (s' - oc'), 
(t - oc) (t' - oc'), 
(u — oc) (u' — a'), 
(v - oc) (V - a'), 
0-/3)(Y-/3'), 
(t -/3)0' -/3'), 
o - /3) O' - /8'), 
0-/3)O'-/S'), 
(5 - 7 ) 0' - 7'), (s - 8) (s' - 8') 
0-7) O'-7'), (t-8)(t'-8') 
(u — 7) (A — 7'), (w—8)(w'— 8') 
0 ~ 7)0' ~ 7O 0 ~ 8) 0' - 8') 
which gives various forms of the equation of homography. In particular, if s = oc, s' = /3', 
t = /3, t' — a', u = 7, v! = 8', v — 8, v = 7, then 
(7 -oc)(8'-oc'), (7 — /3) (8' — ¡3') 
(8 — oc) (7' — oc'), (8-/3) (7'-/3') 
Fd' = 
0-7)(/3'-7'), («-8) 08'-80 
(/3 - 7) (a' - 7'), (/3 - 8) (a' - 8')