524 A FOURTH MEMOIR UPON QUANTICS. [l55 
86. There is another family of covariants which remains to be noticed. Consider 
any two quantics of the same order, 
(a, b, ...\x, y) m , 
{a, b\ ...Jx, y) m , 
and join to these a quantic of the next inferior order, 
(u, v,...\y, —x) m ~ 1 , 
where the coefficients (u, v, ...) are considered as indeterminate, and which may be 
spoken of as the adjoint quantic. 
Take the odd lineo-linear covariants (viz. those which arise from the odd émanants) 
of the two quantics ; the term arising from the (2i + l)th émanants is of the form 
(A, B,....\x, y) 2 «™- 1 - 2 *, 
where {A, B,...) are lineo-linear functions of the coefficients of the two quantics. 
Take also the quadriGOvariants of the adjoint quantic ; the term arising from the 
(2i — m)th émanant is of the form 
(U, V, y)»(«-*-*>, 
where (U, V,...) are quadric functions of the indeterminate coefficients (u, v,...). We 
may then form the quadrinvariant of the two quantics of the order 2 (m — 1 — 2i) : 
this will be an invariant of the two quantics and the adjoint quantic, lineo-linear in 
the coefficients of the two quantics and of the degree 2 in regard to the coefficients 
{u, v,...) of the adjoint quantic; or treating the last-mentioned coefficients as facients, 
the result is a lineo-linear m-ary quadric of the form 
(&, 23, 
viz. in this expression the coefficients 23, ... are lineo linear functions of the co 
efficients of the two quantics. And giving to i the different admissible values, viz. 
from i = 0 to i — \vn — 1 or ^ (m — 1) — 1, according as m is even or odd, the number 
of the functions obtained by the preceding process is fyn or ^ (m — 1), according as 
m is even or odd. The functions in question, the theory of which is altogether due 
to Professor Sylvester, are termed by him Gobezoutiants ; we may therefore say that 
a cobezoutiant is an invariant of two quantics of the same order m, and of an adjoint 
quantic of the next preceding order m — 1, viz. treating the coefficients of the adjoint 
quantic as the facients of the cobezoutiant, the cobezoutiant is an m-ary quadric, the 
coefficients of which are lineo-linear functions of the coefficients of the two quantics, 
and the number of the cobezoutiants is \m or \ {m — 1 ), according as m is even or 
odd. 
87. If the two quantics are the differential coefficients, or first derived functions 
(with respect to the facients) of a single quantic 
(a, y) m ,