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.