MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
103 
or writing instead of (S^, the number in question is 
^ai(a, + l)(o5,+2)(ai+3)-f-|a,(ai + l)a2+|a2(a2+l). 
The integrals of the degrees 1 and 3 give rise to integrals of the degree 4 j and it 
all the composite integrals obtained as above were asyzygetic, we should have 
A4-^ai(ai + l)(ai+2)(ai+3)— ^ai(ai + l)a2— ^a2(a2+l)— a,a3, 
i. e. as the number of irreducible integrals of the degree 4 ; but if there exist any 
further syzygies between the composite integrals, then 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 maybe 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 inde- 
pendent 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 integrals, 
it is easy to see that every other integral will be a rational function of the t'-f-l inte- 
grals ; but every such other integral will not in general be a rational and integral 
function of the v -\- 1 integrals ; and 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.^ is given 
(this will appear in the sequel) as the coefficient of in the development, in ascend- 
ing powers of x, of a rational fraction where fx is of the form 
(1 — xy'{i—x^YK.{] — 'x’^Y’c, 
and the degree of (px is less than that offx. We have therefore 
1 -j- 
and consequently 9a7=(l — x)^‘““'(l — ^^)^ 2 ~" 2 ..( 1 — x*)^*"“*(l— 
Now every rational factor of a binomial X—x^ is the irreducible factor of \—x”'', 
where m! is equal to or a submultiple of m. Hence in order that the series a,, 
may terminate, (px must be made up of factors each of which is the irreducible factor 
of a binomial 1— ■x®, or if (px be itself irreducible, then <px must be the irreducible 
factor of a binomial l—x™. Conversely, if <px be not of the form in question, the 
series &c. will go on ad infinitum, and it is easy to see that there is no point 
in the series such that the terms beyond that point are all of them negative, i. e. there 
will be irreducible covariants or invariants of indefinitely high degrees ; and the 
p 2 
