408 
[696 
696. 
CALCULATION OF THE MINIMUM N.G.F. OF THE BINARY 
SEVENTHIC. 
[From the American Journal of Mathematics, t. II. (1879), pp. 71—84.] 
For the binary seventhic (a, ...\x, y) 7 the number of the asyzygetic covariants 
{a, ...) e (x, yY, or say of the deg-order (6. f), is given as the coefficient of a e x >1 - in 
the function 
1 — ax 7 .1 — act?. 1 — ax 3 .1 — ax. 1 — ax -1 .1 — ax~ 3 .1 — ax~ 5 .1 — ax~ 7 
developed in ascending powers of a. See my “Ninth Memoir on Quantics,” Phil. 
Trans., t. clxi. (1871), pp. 17—50, [462]. 
This function is in fact 
where, developing in ascending powers of a, the second term 
contains only 
negative powers of x, and it may consequently be disregarded: the number of 
asyzygetic covariants of the deg-order (6. y) is thus equal to the coefficient of aV* in 
the function A {x), which function is for this reason called the Numerical Generating 
Function (N.G.F.) of the binary seventhic; and the function A (x) expressed as a 
fraction in its least terms is said to be the minimum N.G.F. 
According to a theorem of Professor Sylvesters (Proc. Royal Soc., t. xxvin. 
(1878), pp. 11—13), this minimum N.G.F. is of the form 
Z 0 + aZ 1 + a 1 2 Z. 2 + ... 4- a M Z 36 
1 — ax. 1 — ax 3 .1 — ax 5 .1 — ax 7 .1 — a 4 . 1 — a 6 .1 — a 8 .1 — a 10 .1 — a 12