VIEWED WITH RELATION TO THE METHOD OF INVARIANTS. 
525 
i being' any integer such that 2/+1 does not exceed m, and now consider 
Eg, + 1.9 ns two functions of the deg’ree in ^ (a?and^ being regarded 
as constants) ; and by virtue of the formula in the last article, form T^, the lineo-linear 
combinant of Egi+i.y and T,- will then be lineo-linear in respect to the 
coefficients in / and (p, and of the degree 2(m-{2i+l)) in respect to ^ and u. 
Again, let 
E,.n= 
1 .2...2i 
m(m — 1) ...{m — 2i + l)‘ 
E,i.CL treated as a function of | and of the degree 2i will furnish a quadrinvariant 
Qi of the degree 2{m—l — 2i) in respect of x and y, and quadratic in respect of the 
system u„ Wg, ... u^. We have thus two forms, and Q^, each of the same even 
degree ( 2 m— ( 2 i+l)) in respect of x, y. Forming between these the lineo-linear 
invariant G„ will be a function lineo-linear in respect of the coefficients of 
/ and <p, and quadratic in respect of the system u^, ... Moreover, G,- will 
(by the general principle of successive concomitance) be an invariant in respect to 
the system /, (p, CL, and cornbinantive in respect to / and <p. Thus then G^ for all 
admissible values of i will belong to the family of forms to which the Bezoutiant is 
to be referred. 
It requires to be noticed, that when ^ is taken ( 0 ), so that and G^ are of the 
degree 2 (m— 1 ), E,. for this case must be taken equal to CL\ which evidently fulfills 
the required conditions of being of the degree 2 (m-l) in {x, y), and quadratic in 
respect of the coefficients of CL. If, now, m be even, we may take for 2/-1-1 suc- 
cessively all the odd numbers from 1 to (m — 1 ) inclusively, and there will be 
m 
2 forms G,.; when m is odd we may take for 2i-\-l successively all the odd numbers 
fiom 1 to m, and the number of forms of G^ will be — It should be observed, that 
when m is odd and 2/-f-l=m, T, will become identical with the lineo-linear combinant 
to/* and p and with the quadrinvariant to CL ; and no power of x or y will enter into 
either, so that G„, will become simply T„,xQ„,. I am now able to enunciate the 
proposition, that G^, G„ ... G^_^ , when m is even, and Go, G„ ... G„,_i, when m is odd, 
2 ~ 
form the constituent scale of forms, of which the Bezoutiant and all other lineo-linear 
quadiatic functions of m variables, which are combinants of the system p, will be 
numerically-linear functions. I propose to term the members of this scale Co-bezou- 
tiants. 
As regards the present memoir, I shall content myself with exhibiting a partial 
verification of this law as regards the connection of the Bezoutiant with the G scale 
of Co-bezoutiants, and a complete determination of the numerical multipliers which 
expiess this connection for the cases comprised between m — 2 and m=6 taken in- 
clusively. It is impossible to predict for what ulterior purposes in the development 
of the Calculus of Invariants these numbers may or may not be required, and it seems 
