AND IMAGINARY ROOTS OF EQUATIONS. 627 
It may be said that the case of three or more equal roots existing in F ( x , y ) has been 
turned into an operator becomes are‘ 
(£)’(: 
which, applied to the Hessian, viz. 3aeu 4 v 2 afu 3 v 3 — eV, after 
multiplying by — -Jg-, gives X— — 2a 3 e 3 , so that D=<F — 128K=a 4 / 4 +256« 3 e 5 , which is capable of easy verifi- 
cation. In fact D becomes the resultant of au 4 -\-ev 4 and v\4eu-\-fv); v 3 introduces the factor a 3 into D ; and 
further, making u:v:: —f: 4e and substituting in au 4 +ev 4 , we obtain the other factor a/ 4 +256e 5 . 
If we adopt u 5 +5euv 5 +v 4 as the reduced form for the failing case (a form analogous to the well-known one, 
w 4 +6cwV+v 4 , for the general quartic), to find e we have J=^ 10 , K= — Hence e 5 = — ; thus when 
K=0, «=0. 
( b ) By a linear transformation we may always take away any two (except the two first or last) coefficients of 
a given quintic, but the vanishing of more than two coefficients always corresponds to some invariantive con- 
dition. Thus, ex. gr., in the form 
ax 3 + 5exy 4 + fy 3 
II 
o 
ax 3 +fif 
L=0 
K=0 
ax 5 + bexy 4 
L=0 
o 
II 
t-5 
ax 3 + 10dx 2 y 3 
J = 0 
K=0 
ax 5 5 bx 4 y + 10 cx?y“ 
L=0 
«-i 
II 
o 
( c ) The condition for the existence of four equal roots in a quintic is the vanishing of the quadratic covariant ; 
that is to say, we must have 
ae— 46cZ+3c 2 =0, af—3be+2cd=0, bf — 4ce + 3cZ' 2 =0. 
The three quantities equated to zero are not separately invariants, but constitute in their ensemble an invarian- 
tive plexus. 
( d ) [It may here be noticed incidentally that the conditions for equal roots in the biquadratic form are 
as follows. For two equal roots, of course, the discriminant is zero, for three equal roots the two lowest in- 
variants are each zero, and for two pairs of equal roots the Hessian (A, B, C, D, Ej£A’, y) 4 becomes to a factor 
prls identical with the primitive (a, b, c, d, ejfcx, y) 4 , so that all the first minors of the matrix 
| a, b, c, d , e, f 
I A, B, C, D, E, F 
vanish. Qucere, whether the character of the five-rayed pencil (centre at origin), in which a, A ; b, B ; c, C ; 
d, D ; e, E mark points, may not serve to distinguish between the case of four real and four imaginary roots.} 
( e ) When J=0 and K=0, but not L=0, it is obvious that p : <r : r : : 1 : < : t 2 , i being any imaginary cube root of 
unity, and the reduced form is m 5 + <i> 5 + ( 2 m> 5 , with the relation u-\-v -\-w=0. 
J and K being zero, D will be so too, and accordingly the equation v? nf' Fw r '—Q will have two equal 
roots. It will easily be found that these equal roots correspond to the system of ratios u=l, v=; 3 , w—i. 
In fact, if we write i/=< 2 +ts, w=< + j 2 p, the equation becomes u 5 -{■ tv 3 + i 2 w 3 =^(30§ +3^ 3 )=0. 
Hence, understanding by a either of the two prime sixth roots of unity, the complete system of ratios of u, v, w 
may be expressed as follows : — 
u = l 
n = l 
u = 1 — -v^lO 
U — 1 -)- v^lOa 
w= 1+ x/lOe 5 
v=i 
v=t" 
v=i 2 — -v^lO 
v=z 4 — Vio 
w=a 4 + v'lOg 
w=i 
w=i 
w=i—P \/10 
w=e 3 -\- v^lOa 5 
w=s 3 — x/10. 
Thus, when J=0 and K=0, u, v, w (with the relation u+v - f w=0) may first be'found, in terms of x, y, by 
MDCCCLXIV. 
4 P 
