A FIFTH MEMOIR UPON QUANTICS.
534
[156
and observing that the equation \ : p= If' : —U implies \U + yU' = 0 = a”(x — 6y) 2 , it
is obvious that the function
(ac-b 2 , ac'-2bb' + ca', a'c - b' 2 \U', - Uf
must be to a factor pres equal to (x — 6,y) 2 (x — Q„yf. But we have identically
(ac — b 2 , ad — 2bb' + ca', a'c' — b' 2 \TJ', — U) 2 = — {(ah' — a'b, ac' — a'c, bd — b’c§x, y) 2 } 2 ,
and we thus see that (x — 6,y), (x — 6 f/ 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, IT; hence we see that the sibiconjugates 6,, 6 y/ of the
involution a, /3, a', /3' are a pair harmonically related to the pair a, /3, and also
harmonically related to the pair a', /3', and this properly might be taken as the
definition for the sibiconjugates 0 /} 6 U of an involution of four terms. And moreover,
a, /3; a!, ¡3' being given, and 9 t , 6 U being determined as the sibiconjugates of the
involution, if cl', (3" be a pair harmonically related to 6Q u , then the three, pairs
a, ¡3; a, /3'; a", ¡3" will form an involution; or what is the same thing, any three
pairs a, /3; ci', ¡3'; a", /3", each of them harmonically related to a pair 6 t , 9 n , will be
an involution, and 6,, 6 y/ 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 6 t , 6 y may be considered as harmonically related to 6 y , 6 /t , and in like manner
6 t/ , 6 it will be harmonically related to 6 y , 6 y/ ; that is, the pairs 6 y , 6 t ; 6 /y , 6 y/ and
a, /3 will form an involution. This comes to saying that the equation
1,
1,
20,
26.
= 0
// > //
1, a + /3, cl/3
is equivalent to the harmonic relation of the pairs cl, /3; 6,, 6 U ; and in fact the deter
minant is
(6, - 0„) (2a£ + 2 e,e„ - (a + /3) (0, + $„)),
which proves the theorem in question.
_108. Before proceeding further, it is proper to consider the equation
1, cl, cl', CLCt! =0,
^ B /3, /3', /3/3'
; B 7> 7 > 77'
1, 5, 8', 88'
which expresses that the sets (a, /3, 7, 8) and (a', /3', 7', 8') are homographie ; for
although the homographie equation may be considered as belonging to the theory of
