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XXX. A Third Memoir upon Quantics. By Arthur Cayley, Esq. 
Received March 13, — Read April 10, 1856. 
My object in the present memoir is chiefly to collect together and put upon record 
various results useful in the theories of the particular quantics to which they relate. 
The tables at the commencement relate to binary quantics, and are a direct sequel 
to the tables in my Second Memoir upon Quantics, vol. cxlvi. (1856) p. 101. The 
definitions and explanations in the next part of the present memoir are given here 
for the sake of convenience, the further development of the subjects to which they 
relate being reserved for another occasion. The remainder of the memoir consists of 
tables and explanations relating to ternary quadrics and cubics. 
Covariant and other Tables, Nos. 27 to 50 (Nos. 1 to 50 binary quantics)*. 
Nos. 27 to 29 are a continuation of the tables relating to the quintic 
{a, b, c, d, e,f\x, yf. 
No. 27 gives the values of the different determinants of the matrix 
a, 
^b. 
6c, 
4d, e ) 
4b, 
6c, 
4d, 
e 
b. 
4c, 
Q)d, 
4e, / 
4c, 
Qd, 
4e, 
/ 
determinants which are represented by 1234, 1235, &c., where the numbers refer to 
the different columns of the matrix. No. 28 gives the values of certain linear 
functions of these determinants, viz. 
L = 1256+ 2345-2.1346 
L' =3.1256— 1346 
8M =—1345 + 2.1246 
8M' = — 2346+2.1356 
8N = — 1245 + 3.1236 
8N' = — 2356+3.1456 
80P = L' — 3L= 5.1346 — 3.2345 
]6P'= — 5L'— L= - 16.1256 — 3.1346 — 2356. 
At the end of the two tables there are given certain relations which exist between 
the terms of Tables 14, 16, 25, 26, 27 and 28. 
* The Tables 49 and 50 were inserted October 6, 1856. — A. C. 
4 o 
MDCCCLVI. 
