veral
the
be
A FIFTH MEMOIR UPON QUANTICS.
where (1), (2), (3) are the quadrics themselves, and (4) is an invariant, linear in the
coefficients of each quadric. And where it is convenient to do so, I write
(1) = U,
(2) = U',
(3) = U",
(4) = n.
105. The equation XI = 0 is, it is clear, the condition to be satisfied by the
coefficients of the three quadrics, in order that there may be a syzygetic relation
\U + fiU' + vU" = 0, or what is the same thing, in order that each quadric may be
an intermediate of the other two quadrics ; or again, in order that the three quadrics
may be in Involution. Expressed in terms of the roots, the relation is
1, a + /3 , a/3
1, a'+/3', otff
T, o" + /3", a"/3'
= 0;
and when this equation is satisfied, the three pairs, or as it is usually expressed, the
six quantities a, /3; cl, /3'; cl", /3", are said to be in involution, or to form an
involution. And the two perfectly arbitrary pairs cl, /3; cl', ¡3' considered as belonging
to such a system, may be spoken of as an involution. If the two terms of a pair
are equal, e.g. if cl” — /3" = 9, then the relation is
1, 2 9 , 9 2
1, CL +/3, (2/3
1, a' + /3', cl' /3'
= 0;
and such a system is sometimes spoken of as an involution of five terms. Con
sidering the pairs (cl, ¡3), (a', /3') as given, there are of course two values of 9 which
satisfy the preceding equation; and calling these 9 / and 9 t/ , then 9 / and 9 U are said
to be the sibiconjugates of the involution cl, /3; a!, ¡3'. It is easy to see that 9 t , 9 0
are the roots of the equation H — 0, where H is the Jacobian of the two quadrics
U and TP whose roots are (a, /3), (a', /3'). In fact, the quadric whose roots are 9 t , 9 U is
y\
2 ycc ,
x 2
1,
a + /3,
a/3
1,
a' + fr
«'/S'
which has been shown to be the Jacobian in question. But this may be made clearer
as follows:—If we imagine that X, /a are determined in such manner that the inter
mediate \U + fjbU' may be a perfect square, then we shall have \U + ¡iTJ' = a"(x — °y)\
where 9 denotes one or other of the sibiconjugates 9 1 , 9 t/ of the involution. Bat the
condition in order that \JJ-V \xTJ' may be a square is
(ac — b 2 , ac —2 bb' + ca', ad — b' 2 l[\, ¡if;