522 
A FOURTH MEMOIR UPON QUANTICS. 
[155 
where J is a number; and then considering the equation x m —1=0, we have to 
determine N the equation 
£(«, ft ...) = (-) m - 1 ^ r . 
But in general 
and if 
then 
£ (a, /3...) = (-) im(m_1) (a - ft (a - 7)... (/3 - a) (/3 - 7)... 
(f)X = (x — a) (x — /3) ...., 
(a-ftK<x-y) ...=<f>'a, &c., 
£(«, ft = f«f/3...; 
<f)x = x m — 1, fix = mx m ~ 1 , 
fi afi /3... = m m (or/3 7 . . .) m_1 , 
(—) m a/3 7 ... = — 1, 
a/3 7 ...= (-y*-* 1, 
fiafi/3... = (-)< w ~ 1 » 2 m m = (-) w ~ 1 m™ ; 
f(a, /3...) = (_)w-i+P«(w-d JV } 
jy — 1) 
a m ~ 2 £(a, /3, ...) = (a 7 ' 1 - 1 a vWl_1 + &c.), 
or what is the same thing, the value of the discriminant □ (= a m_1 a W_1 + &c.) is 
/ 3 J ...). 
It would have been allowable to define the discriminant so as that the leading term 
should be 
1) a rn-i a 'm-i } 
in which case the discriminant would have constantly the same sign as the product 
of the squared differences ; but I have upon the Avhole thought it better to make 
the leading term of the discriminant always positive. 
83. A quantic of an even order 2p has an invariant of peculiar simplicity, viz. 
the determinant the terms of which are the coefficients of the pth differential 
coefficients, or derived functions of the quantic with respect to the facients ; such 
invariant may also be considered as a tantipartite invariant of the j>th émanants. 
Thus the sextic 
or 
here 
and therefore 
but 
or 
and 
whence 
or 
and consequently 
(a, b, c, d, e, f, y\x, y) a