[ 101 ] 
VI. A Second Memoir upon Quantics. By Arthur Cayley, Esq. 
Received April 14, — Read May 24, 1855. 
The present memoir is intended as a continuation of my Introductory Memoir 
upon Quantics, t. 144. (1854) p. 245, and must be read in connexion with it; the 
paragraphs of the two Memoirs are numbered continuously. The special subject of 
the present memoir is the theorem referred to in the Postscript to the Introductory 
Memoir, and the various developments arising thereout in relation to the number 
and form of the covariants of a binary quantic. 
25. I have already spoken of asyzygetic covariants and invariants, and I shall have 
occasion to speak of irreducible covariants and invariants. Considering in general 
a function u determined like a covariant or invariant by means of a system of partial 
differential equations, it will be convenient to explain what is meant by an asyzygetic 
integral and by an irreducible integral. Attending for greater simplicity only to a 
single set (a, h, c...), which in the case of the covariants or invariants of a single 
function will be as before the coefficients or elements of the function, it is assumed 
that the system admits of integrals of the form m=P, m=Q, &c., or as we may 
express it, of integrals P, Q, &c., where P, Q, &c. are rational and integral homogeneous 
functions of the set {a, h, c..), and moreover that the system is such that P, Q, &c. 
being integrals, (p(P, Q..) is also an integral. Then considering only the integrals 
which are rational and integral homogeneous functions of the set (a, b, c..), integrals 
P, Q, R,.. not connected by any linear equation or syzygy (such as XP+jM/Q+>'R--=0*), 
are said to be asyzygetic ; but in speaking of the asyzygetic integrals of a particular 
degree, it is implied that the integrals are a system such that every other integral of 
the same degree can be expressed as a linear function (such as AP+|M/Q+j/R..) 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 Aj, A^, A 3 , &c. denote the number of asyzygetic integrals of the 
degrees 1, 2, 3, &c. respectively, and let Kj, 0 , 3 , &c. be determined by the equations 
Ai = ai 
A 2 =^ai(a,-f l)-l-a 2 
A3= |“i(ai + 1 )(a,+2) +a,052-l-a3 
A 4 =^a,(ai-f-l)(ai-l- 2 )(ai-|- 3 )-|-^o 5 i(o 5 i-l-l)a 2 H-aia 3 -l-^a 2 (a 2 +l)+a 4 , &c., 
* It is hardly necessary to remark, that the multipliers X, y., v and generally any coefficients or quantities 
not expressly stated to contain the set (a, b, c..), are considered as independent of the set, or to use a con- 
venient word, are considered as ‘ trivials.’ 
MDCCCLVI. 
P 
