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;