556 
A FIFTH MEMOIR UPON QUANTICS. 
[156 
In the case of two pairs of equal roots, we have 
a“ 1 U = {x — ay) 2 (x — 7y) 2 , 
a ~ 2 I — tV ( a ~ 7> 4 j 
a 3 J = — (a — 7)®, 
□ = 0, 
a~ 1 H=- t l (a - y) 2 (® - ay) 2 (« - yy) 2 , 
3> = 0; 
these values give also 
eiH - d ju=o. 
146. In the case of three equal roots, we have 
a~ l U—{x — ay) 2 (x — By), 
1=0, J= 0, □ = 0, 
a~ 2 H = —^ (a —B) 2 [2 (x — By) 2 + (x — ay) 2 } (x — ay) 2 , 
a_3<i> = h ( a - 5 ) 3 0 - a y) 6 ; 
and in the case of four equal roots, we have 
a -1 U=(x— ay) 4 , 
1=0, J= 0, □ = 0, 
tf=0, <£ = 0. 
The preceding formulae, for the case of equal roots, agree with the results obtained 
in my memoir on the conditions for the existence of given systems of equalities 
between the roots of an equation. 
Addition, 7th October, 1858. 
Covariant and other Tables (binary quadrics Nos. 25 bis, 29 A, 49 A, and 50 bis). 
Mr Salmon has pointed out to me, that in the Table No. 25 of the simplest 
octinvariant of a binary quintic 1 , the coefficients — 210, —17, +18 and + 38 are 
erroneous, and has communicated to me the corrected values, which I have since 
verified: the terms, with the corrected values of the coefficients, are [shown in the Table] 
No. 25 bis. 
[The terms with the erroneous coefficients were abc 2 d 2 ef ac 5 f 2 , b 4 d 2 f 2 , bc 3 d 3 e ; the 
correct values —220, —27, +22, and +74 of the coefficients are given in the Table 
Q, No. 25, p. 288.] 
1 Second Memoir, Philosophical Transactions, t. cxlvi. (1856) p. 125.