63
PART III
SYMBOLIC NOTATION
The Notation and its Immediate Consequences, §§ 39-41
39. Introduction. The conditions that the binary cubic
(1) / = aoXi 3 +3aiXi 2 ;*:2+3a2XiX2 2 +a3X2 3
shall be a perfect cube
(2) (a 1X1 -\-0i2X2Y
are found by eliminating a\ and «2 between
(3) ai 3 = ao, «i 2 c*2 — 01, cciQ!2 2 = U2, a2 3 =fl3,
and hence the conditions are
(4) aoU2 = ui 2 , aia3 = a2 2 .
Thus only a very special form (1) is a perfect cube.
However, in a symbolic sense * any form (1) can be rep
resented as a cube (2), in which ai and «2 are now mere symbols
such that
(3') ai 3 , ai 2 o;2, «iQ!2 2 , <*2 3
are given the interpretations (3), while any linear combination
of these products, as 2ai 3 — 7oc2 3 , is interpreted to be the cor
responding combination of the a’s, as 2ao~ 7a^. But no inter
pretation is given to a polynomial in on, «2, any one of whose
terms is a product of more than three factors a, or fewer than
three factors a. Thus the first relation (4) does not now follow
from (3), since the expression ai 4 a2 2 (formerly equal to both
* Due to Aronhold and Clebsch, but equivalent to the more complicated
hyperdeterminants of Cayley.