AND IMAGINARY ROOTS OF EQUATIONS. 
619 
Hence its quadratic covariant is the quadratic invariant of 
(jr—t)u—tv, —tu—tv, —tii— tv, — tu—tv , — tu-\-(s— t)v$y!, v'j , 
which is obviously 
— rtu 2 — stv 2 + (rs —rt— st)uv. 
Of this the quadratic invariant is 
rt . st— ^(rs— rt— st) 2 ; 
or writing g=st, a—tr , r—rs, and calling this invariant (I), 
(I) = — V— 2<w— 2*?). 
Again, the cubic covariant or canonizant has been already shown to be rst(ii 2 v-\-uv 2 ). 
Calling the discriminant of this (L), we have 
(L)= - JyrW( 30 )= ygW. 
Again, to find the discriminant (D) in respect to u, v. 
When ru 5 -{-sv 5 -\-tw b = 0 has two equal roots, and w+'y+w=0, it is easy to see that 
we have ru 4 -\-’K= 0, A= 0, tw 4 -\r'k= 0. 
Hence to a constant factor pres (D) will be the Norm of 
{stf -{-(trf +(rs)%, i. e. of g*-f o^+r^( 31 ). 
To find the value of this norm, suppose ^+ o ^+ r ^ = 0, then 
? +<r+r=2 ( ? M+<tV +T* S *), 
and 
§ 2 fi-o- 2 -)-r 2 — 2pv — 2 § r — 2ffr=8 g V® r*( g^ + o' 21 -J- r*) . 
Hence 
(g 2 -b<7 2 -|-T 2 — 2g<r— 2gr— 2<7r) 2 =64g<rr{(gfi-<7-}-r)-}-2(g%^-|-ff if r*-f-‘r 5 g ls )} =128g<7r(gfi- c +v). 
Hence (D) must contain (J) 2 — 128g<rr(g -fir+r) as a factor; and since when £=0, £=0, 
<r=0, and (D)=r 4 =(J) 2 , it is clear that (D) = (J) 2 — 128(K), where 
(K)=g<rr(g+ff+r). 
(39) Although in the investigation in view (K) will only figure as an abbreviation of 
, it may not he amiss to indicate a direct process for finding it. Let us for this 
purpose act upon the Hessian of <b, treated as a function of u, v twice with the canoni- 
zant of O converted into an operator by substituting — ~ in place of u and v. 
(30) p or this is (0, O'jfcu, v) 3 , and the discriminant of (a, b, c, d^u, v) 3 is 
a?d 2 + 4ac 3 + 4 db 3 — 36 V — 6 abed. 
( 31 ) It is worthy of observation that (J) is also a Norm, viz. of gi+tri+ri, so that (J) is the discriminant of 
rv?+sv 3 +tw z . I have not been able to perceive the morphological signiflcancy of this relation. 
MDCCCLXIV. 4 O 
