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1 
155] 
A FOURTH MEMOIR UPON QUANTICS. 
521 
+ 1. It was noticed in the Introductory Memoir, that, by Joachimsthal’s theorem, the 
discriminant, on putting a — 0, becomes divisible by b 2 , and that throwing out this 
factor it is to a numerical factor pres the discriminant of the quantic of the order 
(w—1) obtained by putting a = 0 and throwing out the factor x; and it was also 
remarked, that, this theorem, combined with the general property of invariants, afforded 
a convenient method for the calculation of the discriminant of a quantic when that 
of the order immediately preceding is known. Thus let it be proposed to find the 
discriminant of the cubic 
(a, b, c, yf. 
Imagine the discriminant expanded in powers of the leading coefficient a in the form 
Aa 2 + Ba+ C, 
then this function qua invariant must be reduced to zero by the operation 3bd a + 2cd b + dd c ; 
or putting for shortness V = 2cd b + dd c , the operation is V + 3bd a , and we have 
and consequently 
a-V A +a VH + V(7 "> 
l = 0, 
+ a 6bA + 3bB\ 
B = 
Ivo, A — 
VA= 0. 
But C is equal to b 2 into the discriminant of (35, 3c, d\x, y) 2 , that is, its value is 
b 2 (12bd — 9c 2 ), or throwing out the factor 3, we may write 
(7 = 4<b 3 d — 3 b 2 c 2 \ 
this gives 
B = - ^ (— 6b 2 cd + 24b 2 cd —12be 3 ), 
or reducing 
B = — 6bcd + 4c 3 ; 
and thence 
1 
A + 12c 2 cZ — 12c 2 d), 
or reducing 
A = d 2 , 
which verifies the equation VM=0, and the discriminant is, as we know, 
a 2 d 2 — 6abcd •+ 4ac 3 + 4 b 3 d — 3 b 2 c 2 . 
82. If we consider the quantic (a, b,...a^x, l) m as expressed in terms of the 
roots in the form a (x — ay) (x — /3y)..., then the discriminant (= a m ~ l a m_1 + &c. as 
above) is to a factor pres equal to the product of the squares of the differences of 
the roots, and the factor may be determined as follows: viz. denoting by ¿f(ot, /3, ...) 
the product of the squares of the differences of the roots, we may write 
£ (a, /3, ...) = N (a r ' l ~ l a' m_1 4- &c.), 
C. II. 
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