522 MR. SYLVESTER ON THE FORMAL PROPERTIES OF THE BEZOUTIANT 
linear quadratic covariants of m variables to every two homogeneous functions of and 
y of the mth degree. I have, moreover, I believe, succeeded in determining the number 
of such lineo-linear quadratic forms for any value of {m), of which all the rest, in 
whatever manner obtained, may be expressed as linear functions, the coeflacients of the 
linear relations moreover being abstract numbers ; m other words, I have succeeded in 
forming the fundamental or constituent scale of lineo-linear quadratic forms of m vari- 
ables covariantive to /and ; a result of too great interest, as exhibiting the affinities of 
the Bezoutiant to its cognate forms, to be altogether passed over in silence. Supposing 
the number of linearly independent forms of the kind to be v, then speaking a 
any of the forms taken at random might seem to be equally eligible to form one of 
the V included in the fundamental scale, combined with any (v— 1) others independent 
inter se, and of which the selected one is also independent. In fact, however, this is 
not so ; for it will always be more satisfactory to contemplate the fundamental scale of 
forms as generated successively or simultaneously by a uniform process ; and in the 
case before us, the process which I have hit upon, and which I believe is the simplest that 
can be employed for generating the fundamental scale, will be found not to include 
directly the Bezoutiant among the number. There will thus arise two subjects of 
inquiry ; 1st, the mode of forming the fundamental scale, and proving its fundamental 
D=0 is the condition to be satisfied in order that J may be representable imder the form of the sum of the 
squares of (m-1) linear functions of ^ and y, and D itself is an invariant to J, and consequently an invanant 
and (as is obvious from its form) a combinantive invariant to / and Moreover, ^hich is more immediate y 
to the point, vs^e know that the quadratic form Q 
^ Aom! + 2A,(u , . (m - 1 )«,) + A,|( (>» - ^>,7 + Waj +&<=• + ^ 2 , 
will be an invariant to /, f and (this last quantity being defined as inp. 524), and a combinantive covanant 
to/ and 0 in the same sense precisely as the Bezoutiant is a covariant to the same, and hke the Bezoutiant 
is lineo-linear in respect of the coefficients off and If we operate with the symbol E, where E represents 
^^(ul + 2a,.«3) + &c. 
upon K any invariant of /and <p, we shall obtain ^l.K, a quadratic function of v,v....v,n. which by the rules of the 
Calculus of Forms we knowwill be a contravariant to/ and <p, and the matrix coiuesponding to which must evi ent y 
be persymmetrical. It is an interesting subject of inquiry, which I reserve for some future occasion, to determine 
the Co-bezoutiant, the Discriminant of which must be employed for K, so that when this discnmmaut is operatea 
upon by E, the matrix corresponding to E .K may become identical (term for term) with the matrix which is 
the inverse to the Bezoutiant matrix, which inverse, as Jacobi has so simply and beautifully demon.tia 
possesses this persymmetrical character. Vide the “ De Eliminatione,” section 5._ The investigation of the 
arithmetical connexion between the Q of this note and the fundamental Co-bezoutiants must be also simi a^r j 
reserved. I believe it to be generally true, and have verified the fact for the case of two^ cubic functions, that 
E.Q. gives a quadratic form such that the corresponding matrix is the inverse to the matrix of Q. Ihe calcu- 
lations necessary for extending the verification of this remarkable proposition for functions of a-, y exceedmg 
the third degree (notwithstanding that they are much abbreviated by the application of the rules of t le c - 
cuius) still remain excessively laborious. The abbreviation alluded to consists in confining the verification in 
question to the comparison of either one of the two unreiterated terms at opposite corners of the matrix to 
£.Q with the corresponding term in the inverse matrix of Q ; if these coincide, it is easy to prove that eierv 
other pair of corresponding terms in the two matrices must also coincide respectively with one another. 
