102 
MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
or what is the same thing-, suppose that 
1 +Ai>r+A 2 a;^+ &c. = (l — 
A little consideration will show that 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. Ai=a,. Represent for a moment the 
irreducible integrals of the degree 1 by X, X', &c., then the composite integrals 
X^, XX', &c., the number of which is must be included among the asyzy- 
getic integrals of the degree 2 ; and if the composite integrals in question were asyzy- 
getic, there would remain A2—^ai(o5i-f 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 Ag will be less the number of these 
syzygies, and we shall have Ag — ^ai(ai + l), e. equal to the number of the irre- 
ducible integrals of the degree 2 less the number of syzygies between the composite 
integrals of the same degree. Again, suppose that a,^ is negative =—(3^, we may for 
simplicity ouppose that there are no irreducible integrals of the degree 2, but that 
the composite integrals of this degree, X^ XX', &c., are connected by (3^ syzygies, 
such as &c.=0, &c.=0. The asyzygetic integrals of 
I 
the degree 4 include X^, X®X', &c., the number of which is ^ai(ai + l)(«i + 2)(aiH-3 ); 
but these composite integrals are not asyzygetic, they are connected by syzygies which 
are augmentatives of the jSa syzygies of the second degree, viz. by syzygies such as 
(xX^-l-|a-XX'..)X^=0, (?vX^+^XX'..)XX'=0 &c. (;v,X^-l-,w,iXX'..)X^=0, 
(X,X^+iM'iXX'..)XX'=:0, &c., 
the number of which is|ai(ai-l- 1 )^ 2 - And these syzygies are themselves not asyzy- 
getic, they are connected by secondary syzygies such as 
-k(X,X^+f^,XX'.,)X^-f^(X,X^+f^,XX'..)XX'-kc.=0, See. Sec., 
the number of which is^j32(/32— 1). The real number of syzygies between the com- 
posite integrals X^, X^X', &c. (i. e. of the syzygies arising out of the syzygies 
between X^, XX', &c.), is therefore |a,(ai-j-l)j32— ^(32(|32— 1), and the number of inte- 
grals of the degree 4, arising out of the integrals and syzygies of the degrees 1 and 2 
respectively, is therefore 
1)(«i+2)(c 4,-1-3) — 2ai(a,-[- l)^32+2^2(f32— 1) ; 
