622 
PEOEESSOE SYLVESTEE ON THE EEAL 
positive, or to speak more strictly non-negative, we have 
G = JK 4 + 8LK 3 — 2 J 2 LK 2 — 7 2 JL 2 K — 4 32L 3 + J 3 L 2 ( 35 ). 
It is evident that G must be identical to a positive numerical factor pres with the 
function which M. Hermite denotes by I 2 ( 36 ). 
( 3S ) It will be observed tbat when J=0 and L=0, G vanishes. This is easily verifiable a, priori-, for when 
J=0 and L=0, the reduced form, has been seen to be ax s +5exy 4 , of which the canonizant is 
a 0 0 0 
0 0 0 e 
0 0 e 0 
y 3 — y 2 x yx 2 — x 3 
which equals aexy 2 . 
Hence the form and its canonizant have a common factor x, and consequently their resultant vanishes ; 
hence 1=0 and G=I 2 =0. G also vanishes when K=0 and L=0, which is also easily verifiable; for then 
the reduced form becomes u 5 + v s , of which the canonizant vanishes, and consequently the resultant of the form 
and its canonizant becomes intensely zero ; which accounts for the high power of K in (JK 4 ), the sole term of 
G in which L does not appear. 
(36) Compare expression for 16I 2 , Cambridge and Dublin Journal, p. 203. This will be found to contain 
nine terms, and to rise as high as the fifth power in A (which to a constant factor pres is identical with my J); 
whereas in there are only six terms, and no power of J beyond the third. This seems to indicate that the 
K and L are more fortunately chosen than M. Hermite’s J 2 , J 3 , which are invariants of the like degrees 8 and 
12. It is of course evident that the following relations exist between M. Hermite’s A 1? J 2 , J 3 and the J, K, L 
of this paper, 
J 2 =mP +nK, 
J 3 =pJ 3 + q JK + rL, 
where Z, m, n, p, q, r are certain numerical quantities. Until these are ascertained, it is impossible to con- 
front H. Hermite’s results with my own, to ascertain whether or not they are identical in substance, and, if 
not, wherein the difference consists. I therefore subjoin the necessary calculations for effecting this important 
object. 
Let us first take the form x 3 +5exy i +y 3 . The quadratic covariant of this is x(ex+y). 
Accordingly, to obtain M. Hebmite’s A, B, C, C', B', A' (Cambridge and Dublin Journal, vol. ix. p. 179), we 
must make 
x—JL ; ex+y—Y, 
which gives (vide C. and D. J. p. 180) 
E =X 5 + 5eX(Y— eX) A + (Y-eX) 5 
= (A, B, C, C', B', A'JX, Y) 5 , 
where 
A=l+4e 5 , B= — 3e 4 , C=2e 3 , C'=—e 2 , B'=0, A'=l. 
Accordingly (vide C. and D. J. p. 184), 
AA'-3BB'+ 2CC'=l + 4e 5 - 4e 5 =l = Va, 
AA'+ BB'- 2CC'=l+4e 5 + 4e 5 =l + 
8e 5 = 
JL_ 
2VA 3 ’ 
A A' + 5BB' + 10CC' = 1 + 4e 5 — 20e 5 = 1 - 1 6e 5 
Hence 
I 2 
= 2VA S 
A=l, I l= 2+16e s , I 3 =2— 32e 5 . 
Again (vide C. and D. J. p. 186. § vii.), 
8 J x = Ij — A 2 = 1 + 1 6e 5 , 24J 2 =I 2 — 2I l A + A 3 =-l-64e 5 ; 
