282 
MR. A. CAYLEY’S SE^TOTH MEMOIR ON QUANTICS. 
(so that T'=A®4'B^ <^11 til foi’ the canonical form, 
A=-4/+4^^ B=l + 8^^). 
Making this change, and joining to the equation inp that derived from it by writing 
2 for and interchanging A, B, we have the three equations 
(1, 0, -6A, -8T , -3A^ 1)^=0, 
(1, 0, -6B , -8T , -3B^ 1)^=0, 
(12A, 8T, -6A^ -6TA, A 1)^=0. 
240. The signification of the equation in is as follows, viz. if the quantic 
u=(*x^% 
is transformed into the canonical form 
X*+Y=’+Z^+6DiYZ 
by means of the linear eqTiations 
{x,y,z)={AX^, Y, Z), 
where A is a matrix, then using the same letter A to represent the determinant formed 
out of this matrix, or determinant of substitution, we have 
3 
so that the equation in q is one that presents itself in the question of the reduction of 
the cubic to its canonical form. 
In fact the linear transformation gives 
TA®= 
and thence 
(P-64S=’)A^=(l + 8/7, 
which, wiiting B^ in the place of T^— 64S^ becomes 
B^A'^= (1 + 8^7, or 
B A^ = 1 + 8^^ , or 8^®=BA^-1, 
whence also 
8TA®= 8— 20(BA*~1)-(BA^-1)‘'' ' 
or, as this may be written. 
=27-18 BA^ - B^A«, 
81 54B 
A8~ A4 
24T 
A2 
3B^=0, 
which, putting therein qz=~^, 
( 1 , 0 , 
becomes 
-6B, -8T, -3B^^^, l/=0. 
the above-mentioned equation in q. 
241. The relation between & and q is 
T , 
