■P 
155] A FOURTH MEMOIR UPON QUANT1CS. 523 
lias for one of its invariants, the determinant 
a, b. c, d . 
c, d, e, f 
d, e, f, g 
The invariant in question is termed by Professor Sylvester the Catalecticant. 
84. Professor Sylvester also remarked, that we may from the catalecticant form 
a function containing an indeterminate quantity X, such that the coefficients of the 
different powers of X are invariants of the quantic ; thus for the sextic, the function 
in question is 
t , b , c , d — X 
d - IX, 
d + hX, 
f 
where the law of formation is manifest; the terms in the sinister diagonal are 
modified by annexing to their numerical submultiples of X with the signs + and — 
alternately, and in which the multipliers are the reciprocals of the binomial coefficients. 
The function so obtained is termed the Lambdaic. 
85. If we consider a quantic of an odd order, and form the catalecticant of the 
penultimate émanant, we have the covariant termed the Canonisant. Thus in the case 
of the quintic 
the canonisant is 
which is equivalent to 
(a, b, c 
d, e, 
/5 
ax 
+ h. 
bx + 
bx 
+ W, 
cx -f 
dy, 
cx 
+ dy, 
dx+ 
ey, 
f, ~ 
ifx, 
yx 1 , 
a, 
b , 
c 
b, 
c , 
d 
c , 
d , 
e , 
and a like transformation exists with respect to the canonisant of a quantic of any 
odd order whatever. The canonisant and the lambdaic (which includes of course the 
catalecticant) form the basis of Professor Sylvester’s theory of the Canonical Forms 
of quantics of an odd and an even order respectively. 
66—2