514 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
different and less simple form ; two such sets of criteria are obtained, and the identifica- 
tion of these, and of a third set resulting from a separate investigation, with the criteria 
of Professor Sylvester, is a point made out in the present memoir. The theory is also 
given of the canonical form which is the mechanism by which M. Hermite’s investiga- 
tions were carried on. The Memoir contains other investigations and formulae in relation 
to the binary quintic ; and as part of the foregoing theory of the determination of the 
character of an equation, I was led to consider the question of the imaginary linear trans- 
formations which give rise to a real equation : this is discussed in the concluding articles 
of the memoir, and in an Annex I have given a somewhat singular analytical theorem 
arising thereout. 
The paragraphs and Tables are numbered consecutively with those of my former 
Memoirs on Quantics. I notice that in the Second Memoir, p. 126, we should have 
No. 26=(No. 19) 2 — 128 (No. 25), viz. the coefficient of the last term is 128 instead of 
1152. 
Article Nos. 251 to 254. — The Binary Quintic, Covariants and Syzygies of the degree 6. 
251. The number of asyzygetic covariants of any degree is obtained as in my Second 
Memoir on Quantics, Philosophical Transactions, t. 146 (1856), pp. 101-126, viz. by 
developing the function 
1 
(1 -*)(1 — aw)( 1 -tfV) (1 —a?z)(\—x i z ) (1 —x b z) ’ 
as shown p. 114, and then subtracting from each coefficient that which immediately 
precedes it ; or, what is the same thing, by developing the function 
1 —x 
(1 —z ) (1 — xz){\ —x 2 z){\—sc i z ) (1 — x 4 z ) (1 — x h z ) 5 
which would lead directly to the second of the two Tables which are there given ; the 
Table is there calculated only up to z 5 , but I have since continued it up to z 18 , so as to 
show the number of the asyzygetic covariants of every order in the variables up to the 
degree 18 in the coefficients, being the degree of the skew invariant, the highest of the 
irreducible invariants of the quintic. The Table is, for greater convenience, arranged 
in a different form, as follows : — 
