MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
91 
system of Aq and Aj regarded as binary ground-forms in 2 ^,^, — as variables.^ To 
this we shall return (§ 34). 
A Special Series of Invariants. 
16. There is a succession of invariants of consecutive orders, comparatively simple 
in form, which can be derived by using the remark made in § 9. A set of invariants 
of the form suggested by the covariants of a binary quantic, which involve only 
differential coefficients of the quantic with respect to the variables, is derivable by 
considering the functions in 2 analogous to Hermite’s “associated covariants'’ which 
may be taken to be 
A„j_2 — 0’ — 1’ —2i .’ ^2, —2’ IJ ^0i ^io) 
for values 2, 3, 4, . . . of m. 
It is easy to see that each of these functions satisfies the equations (v.) and (vi.), 
viz., = 0 and 0 ; these, in fact, are the equations which suggest the 
functions. 
But, when we consider the operators and Ao which do not arise in connexion 
with covariants of binary forms, we have 
A 2 ^ A „;_2 
- (m 1) [\m-l “b 2^2, m — z “b 3-2, ni — 'ipT T 
+ . . . } 
01 -^10 
Ply U 
= (?n — 1) 1,0 — (m — 1) ^ wz, 
— m{m— 1) ZioA«,_3 ; 
and, similarly, 
A„ 
0 py 
0/ 
— Z -l) —3" 
Again, in regard to the operators which occur in (i.) and (ii.), it is easy to show that 
— 3r/(A,„_o, 
3'J7i/A,„_2. 
If, then, we can obtain combinations of A^,, Aj^, Ao, . . . which are homogeneous and of 
uniform grade, such as to satisfy A^f = 0 and Ao/ = 0, these combinations will be 
invariants ; and it follows from the effect of the linear operators Aj and Ao on the 
quantities A that any combination of the A’s which satisfies Af'= 0 will also satisfy 
Ao/- 0. 
* Salmon, ‘ Higlier Algebra’ (3rd edition), § 198; Clebsch, ‘ Tlieorie der bintiren Formen,’ § 59; 
Gordan, ‘ Vorlesungen uber luvariantentbeorie,’ vol. 2, § 31. The quantities A^ A,,^ Lo, Hq 
are, save as to numerical factors, respectively tbe same as Salmon’s symbols «, v, (1,1), L., A, Lj, (2, 0) ; 
as Clebsch’s symbols 0,/, d, r/, D, p, A ; as Gordan’.s symbols 0, /, d, q, A^y; p, A. 
N 2 
