358 PROFESSOR A. R. FORSYTH ON THE DIFFERENTIAL INVARIANTS 
We therefhre require an algebraically complete aggregate of this set of invariants and 
covariants. 
It IS to be noticed tliat the argument V does not appear in the equations (I.,)". (I,)'. 
(Ij.) ; so that it is a solution of the equations, and it must be associated with the 
required algebraically complete aggregate of concomitants of the binary forms. 
The three equations (L)", (I 3 )', (IJ constitute a complete Jacobian system, and the 
number of arguments which occur is 33; hence the algebraicaUy complete aggregate 
of solutions contains 30 Solutions, which thus give the algebraically complete aggregate 
of concomitants of the system of binary forms. 
This aggregate is known=^ to be (or to be equivalent to) the following 
the linear quantic, vtq ; 
tlie quadratic itq, and its Hessian (discriminant) EG — F~; 
• • • • .LN - MG 
to'".,, 
. n c — ; 
. a'c' - lj'~ 
the cubic w.^, its Hessian, and either its discriminant or its cubicovariant; 
the cubic ir'g, its Hessian, and either its discriminant or its cubicovariant; 
the cubic u' g, its Hessian, and either its discriminant or its cubicovariant; 
the quartic its Hessian, its quadrinvariant. and its cubinvariant; 
together with the Jacobians of any one of the forms 10 , say tv.,, with all the rest of the 
forms. This makes up the I'equisite total of 30. 
1 he asyzygetic aggregate is, of course, vastly more extensive; but for the jiresent 
purpose it is only an algebraically independent aggregate that is wanted. Many 
modifications in the latter are possible : what is necessary to secure is that anv 
modification does not interfere with the algebraical completeness of the aggregate. 
For instance, consider the set composed of 
where 
V,, EG — Fy w".,, ac — h^, J (to.., to".,}, 
4J (an, wG) = 
dw., dw"., can cto".. 
10 
in the asyzygetic system, there is an intermediate invariant Ec — '2Fh + Go ; we 
have 
J3 = w.,w"., (Ec - 2F0 + Go) - w,~ (ac - G) - w"F (EG - F-), 
and therefore, in the algebraical aggregate, Ec - 2Fb + Ga can be included when 
any other (such as ac — b~) is excluded. Such a change would be desirable if 
differential invariants, linear in the C|uantities o, b, c, were required. 
See a memoir by the author, ‘American Journal of Mathematics,’ vol. 12 (1890), §§ 17, 22, 30. 
