PEOFESSOE CAYLEY’S NINTH MEMOIE ON QUANT ICS. 
19 
2°, if P, Q, E, ... are covariants, any rational and integral function F(P, Q, E ,. . 
homogeneous in regard to the coefficients, is also a covariant, — we say that the cova- 
riants X, Y, ... of the same degree in regard to the coefficients, and not connected by 
any identical equation aX+(3Y . . . = 0 (where a, (3, . . . are quantities independent of 
coefficients (a, b, c, . . .)), are asyzygetic covariants, and that a covariant not expressible 
as a rational and integral function of covariants of lower degrees is an irreducible cova- 
riant ; and it is assumed that we know the number of the asyzygetic covariants of the 
degrees 1, 2, 3. . . . ; say, these are A„ A 2 , A 3 , . . ., or, what is the same thing, that the 
number of the asyzygetic covariants of the degree d, or form (a, b, . . ./, is equal to the 
coefficient of a 9 in a given function 
<p(tq) 1 -p Aj ci -p A 2 «" . . . — p A gCi 9 -p . . ., 
where I have purposely written a, as a representative of the coefficients (a, b, c , . . .), in 
place of the x of my Second Memoir. 
329. The theory was, that determining «„ « 2 , ... by the conditions 
A ! = «„ 
A 2 — 2°q(°q -p 1) -p 
+ l)( a i + 2)-P«i«2“T a 3-> 
that is, throwing 
into the form 
1 -p Aj< 3! — p A 2 (Z" _ p A 3 ® 3 -p . . . 
(1 — «)~ ai (l — « 2 )~“ 2 (1 — a 3 ) - " 3 . . ., 
the index would express the number of irreducible covariants of the degree r less the 
number of the (irreducible) linear relations, or syzygies, between the composite or noil- 
irreducible covariants of the same degree. Thus A 1 = a 1 , there would be oq covariants 
of the degree 1*; these give rise to ^oq(oq-pl) composite covariants of the degree 2; 
or, assuming that these are connected by Jc 2 syzygies, the number of asyzygetic com- 
posite covariants of the degree 2 would be ^oq(oq + 1) — Jc 2 \ and thence there would be 
A 2 — 2 a i( a i + l )+^25 that is, oq~p Jc 2 irreducible covariants of the same degree; so that 
(irreducible invariants less syzygies) (a 2 +/t' 2 ) — Tc 2 is =« 2 . 
330. The syzygies are here irreducible syzygies; for, calling P, Q, E, . . . the covari- 
ants of the degree 1, there is no identical relations between the terms P 2 , Q 2 , PQ, . . . ; 
imagine for a moment that we could have l 2 such identical relations (viz. this might very 
well be the case if instead of the i \ k 1 (cq-pl) functions P 2 , Q 2 , PQ, . . ., we were dealing 
with the same number of other quadric functions of these quantities), that is, relations 
not establishing any relation between P 2 , Q 2 , PQ, . . . , and besides these non-identical 
relations as above; then the number of irreducible invariants would be a. 2 -\-k 2 -\-l. 2 , and 
the number of irreducible syzygies being as before the difference would be not a 2 
* For the case of covariants, cc 1 is of course =1 ; but in the investigation the term covariant properly stands 
for any function satisfying the conditions 1° and 2°. 
D 2 
