1870.] 



Prof. Caylcy on Qualities. 



313 



different observer. I cannot help entertaining the hope that something of 

 the sort will sooner or later be undertaken with regard to the investigation 

 of the whole nervous system." 



May 19, 1870. 



General Sir EDWARD SABINE, K.C.B., President, in the Chair. 

 The following communications were read 



I. " A Ninth Memoir on Quantics." By Prof. Cayley, F.R.S. 

 Received April 7, 1870. 



(Abstract.) 



It was shown not long ago by Prof. Gordan that the number of the 

 irreducible covariants of a binary quantic of any order is finite (see his 

 memoir " Beweis das jede Covariante und Invariante einer binaren Form 

 eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl 

 solcher Formen ist," Crelle, t. 69 (186i>), Memoir dated 8 June 1868), 

 and in particular that for a binary quantic the number of irreducible co- 

 variants (including the quantic and the invariants) is = 23, and that for a 

 binary sextic the number is = 26. From the theory given in my " Second 

 Memoir on Quantics," Phil. Trans. 1856, I derived the conclusion, which 

 as it now appears was erroneous, that for a binary quintic the number of 

 irreducible covariants was infinite. The theory requires, in fact, a modifi- 

 cation, by reason that certain linear relations, which I had assumed to be 

 independent, are really not independent, but, on the contrary, linearly 

 connected together : the interconnexion in question does not occur in 

 regard to the quadric, cubic, or quartic ; and for these cases respectively 

 the theory is true as it stands ; for the quintic the interconnexion first 

 presents itself in regard to the degree 8 in the coefficients, and order 14 in 

 the variables ; viz. the theory gives correctly the number of covariants of 

 any degree not exceeding 7, and also those of the degree 8, and order 

 less than 14 ; but for the order 14 the theory as it stands gives a non- 

 existent irreducible covariant (a, . .y{x,y) li ; viz. we have, according to 

 the theory, 5 = (10 — 6) + 1, that is, of the form in question there are 10 

 composite covariants connected by 6 syzygies, and therefore equivalent to 

 10 — 6, = 4 asyzygetic covariants ; but the number of asyzygetic covariants 

 being = 5, there is left, according to the theory, 1 irreducible covariant of 

 the form in question. The fact is that the 6 syzygies beir,g interconnected 

 and equivalent to 5 independent syzygies only, the composite covariants 

 are equivalent to 10 — 5, = 5, the full number of the asjzygetie covariants. 

 And similarly the theory as it stands gives a non-existent irreducible co- 

 variant (a, . *Y(m s y) 2 °» The theory being thus in error, by reason that i: 



