252 
A SECOND MEMOIR UPON QUANTICS. 
[141 
the number of which is |/3 2 (/3 2 — 1). The real number of syzygies between the com 
posite integrals X 4 , X 3 X', &c. (i.e. of the syzygies arising out of the ¡3. 2 syzygies 
between X 2 , XX', &c.), is therefore \ ol 2 (a 4 + l)/3 2 — -|/3 2 (/3 2 — 1), and the number of 
integrals of the degree 4, arising out of the integrals and syzygies of the degrees 
1 and 2 respectively, is therefore 
24 a i ( a i +1) ( a i + 2) (a x + 3) — ^ (a 4 + 1) /3 2 +1 ¡3. 2 (¡3 2 — 1); 
or writing — a 2 instead of ¡3 2 , the number in question is 
■24 a i(®i + l)( a i + 2)(a! + 3) + ^ ^(«i + 1) a 2 + ^ a 2 (a 2 + 1). 
The integrals of the degrees 1 and 3 give rise to oqa 3 integrals of the degree 4; and if 
all the composite integrals obtained as above were asyzygetic, we should have 
X 4 - ®i( a i + l)(«i + 2)(a! + 3)-i «!(<*, + l)a 2 - i 3f 2 (a 2 + 1) - 
i. e. a 4 as the number of irreducible integrals of the degree 4; but if there exist any 
further syzygies between the composite integrals, then a 4 will be the number of the 
irreducible integrals of the degree 4 less the number of such further syzygies, and the 
like reasoning is in all cases applicable. 
27. It may be remarked, that for any given partial differential equation, or system 
of such equations, there will be always a finite number v such that given v independent 
integrals every other integral is a function (in general an irrational function only 
expressible as the root of an equation) of the v independent integrals; and if to these 
integrals we join a single other integral not a rational function of the v integrals, it is 
easy to see that every other integral will be a rational function of the v + 1 integrals; 
but every such other integral will not in general be a rational and integral function of 
the v + 1 integrals; and [incorrect] there is not in general any finite number whatever 
of integrals, such that every other integral is a rational and integral function of these 
integrals, i.e. the number of irreducible integrals is in general infinite; and it would seem 
that this is in fact the case in the theory of covariants. 
28. In the case of the covariants, or the invariants of a binary quantic, A n is given 
(this will appear in the sequel) as the coefficient of x n in the development, in ascending 
powers of x, of a rational fraction , where fx is of the form 
jx 
(1 — #)^ x (l — ic 2 )^ 2 ...(l — aXf k , 
and the degree of <f>x is less than that of fx. We have therefore 
and consequently 
1 -f A-2X + A 2 x 2 + 
cf>x = ( 1 -xf 1 ~ ai (l-x 2 f i ~ a \..(l-x k Ÿ k ~ ak 0 -X k + l )~ ah+l ....