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A SECOND MEMOIR UPON QUANTICS. 
251 
the same degree can be expressed as a linear function (such as \P + /jlQ + vM...) of 
these integrals; and any integral P not expressible as a rational and integral homo 
geneous function of integrals of inferior degrees is said to be an irreducible integral. 
26. Suppose now that A 1} A. 2 , A 3> &c. denote the number of asyzygetic integrals 
ot the degrees 1, 2, 3, &c. respectively, and let a. 1 , a 2 , a 3 , &c. be determined by the 
equations 
A\ — &i, 
A 2 = i «i («i + 1) + a 2 , 
A 3 = ^a 1 («! + 1) («! + 2) + a 2 a 2 + or 3 , 
A 4 = ^4 ctj (sq + 1) (ofj + 2) (otj + 3) + ^ a 2 (a 2 + 1) a 2 + a x a 3 + \ a 2 (a 2 + 1) + a 4 , &c., 
or what is the same thing, suppose that 
1 + A& + A^ + &c. = (1 - x)~ ai (1 - x 2 )~ a ' (1 - a?)~ a \.. ; 
a little consideration will show that a r represents the number of irreducible integrals 
of the degree r less the number of linear relations or syzygies between the composite 
or non-irreducible integrals of the same degree. In fact the asyzygetic integrals of 
the degree 1 are necessarily irreducible, i.e. A 2 = a 2 . Represent for a moment the 
irreducible integrals of the degree 1 by X, X', &c., then the composite integrals 
X 2 , XX', &c., the number of which is \ a 1 (ct 1 + 1), must be included among the asyzygetic 
integrals of the degree 2; and if the composite integrals in question were asyzygetic, 
there would remain A 2 — ^ a 2 (a 2 + 1) for the number of irreducible integrals of the 
degree 2 ; but if there exist syzygies between the composite integrals in question, the 
number to be subtracted from A 2 will be less the number of these syzygies, 
and we shall have A 2 — | a 2 (a 2 +1), i.e. or 2 equal to the number of the irreducible 
integrals of the degree 2 less the number of syzygies between the composite integrals 
of the same degree. Again, suppose that a 2 is negative = — /3 2 , we may for simplicity 
suppose that there are no irreducible integrals of the degree 2, but that the com 
posite integrals of this degree, X 2 , XX', &c., are connected by /3. 2 syzygies, such as 
XX 2 + ¡lXX’ + &c. = 0, XjX 2 + /XjXX' + &c. = 0. The asyzygetic integrals of the degree 4 
include X 4 , X 3 X', &c., the number of which is («i + 1) (<*i + 2) (y 3 + 3); but these 
composite integrals are not asyzygetic, they are connected by syzygies which are 
augmentatives of the (3 2 syzygies of the second degree, viz. by syzygies such as 
(XX 2 + /xXX'...)X 2 = 0, (XX 2 + /iXX'...)XX' = 0, &c. (\ 1 X 2 + y a 1 XX'...)X 2 = 0, 
(X 1 X 2 + / a 1 XX'...)XX , = 0, &c., 
the number of which is ^^(otj + l)/^. And these syzygies are themselves not asyzygetic, 
they are connected by secondary syzygies such as 
Xj (XX 2 + fxXX'...) X 2 + fi, (XX 2 -l- fiXX'. ,.)XX'+k 
- X (X,X 2 + frXX'...) X 2 - ya (X,X 2 + fHXX'...) XX' - &c. = 0, &c. &c., 
32—2