628 
PROFESSOR SYLVESTER ON THE REAL 
lost sight of ; but we know, and it is capable of immediate verification by taking as the 
solving the cubic equation, obtained by equating to zero the canonizant of (a, b, c, d, e,f^oc, y), and then x, y 
will be known from the above system of values for any two of the quantities u, v, w. 
0 It is obvious that the form ax’ + dx 2 y 3 gives J=0andK=0 ; but it seems desirable to prove the converse, 
viz. tbatwben J=0 and K=0, but not L=0, the form is always reducible to aw 5 + 10 Su 2 v B , which maybe done 
as follows. Since J=0 and K=0 the discriminant is zero, and we may assume 
E = ax' + 5bx 4 y + 10 cafiy 1 + 10 daPy 3 , 
and we have J = discriminant of 
Hence 
( - 4 bd + 3 c 2 )f + 2cd£y) + 3d 2 rj 2 . 
3d\3c 2 — 450 — <?d ? = 0 ; 
d cannot be zero, for then we should have J =0, K=0, L=0, contrary to hypothesis. Hence 8c 2 — 12bd= 0. 
* . 2c 2 
If 6=0 and c=0, P is already reduced to the desired form; but if not, d = — , and P becomes 
j ’ * 35 
aoc‘ + 
5b 
or, making 
Gx2 y 
5b 
a g = 
, 12c 2 . 8c 2 3 
+- V *f+ W y 3 
x \ ~ c n . 
X + -. 
P =x r> + 10 x 3 v 3 , as was to be shown. 
The corresponding converses for the case of J = 0, L=0, and of K=0, L=0 have been already established. 
( s ) It will be observed that under a certain point of view L for binary quintics is the analogue of A the discri- 
minant for binary quartics, the condition of failure in the general reduced form in the two cases being L= 0 
and A = 0 respectively. The mere vanishing of the discriminant in the case of the quintic function, unattended 
by any other condition, does not affect the nature of the reduced form. 
( h ) It has been shown previously in the text that when L=0 the primitive is reducible to the form 
(a, 0, 0, 0, c, fyjv , y) 5 . 
Hence if I 12 is any duodecimal invariant which vanishes when 5=0, e=0, d— 0, I ]2 must vanish whenever L 
vanishes, and consequently, since L is of as high a degree as I 12 , I 12 must be a numerical multiple of L. In 
Mr. Cavley’s Third Memoir on Quintics, “ No. 29 ” represents a duodecimal invariant calculated by M. Fa! 
de Bntnsro, and characterized morphologically by Mr. Cayley as being that duodecimal invariant in which “ the 
leading coefficient a does not rise above the fourth degree.” On examining No. 29 it will be found to contain 
no term in which 5, c, d are all simultaneously absent. Hence it is, by virtue of the above observation, a mul- 
tiple of my L : to determine the numerical factor, let all the coefficients in the primitive except a , d be supposed 
zero ; then the canonizant becomes 
a 0 0 d 
—d?y z -\-ad?a?. 
y* —y*x yaf —x 6 
Hence L becomes —27 a 2 cl w , but “No. 29” becomes 27 a?d}°. Hence we have the important relation 
“ No. 29” = — L, so that No. 29 is a discriminant, an intrinsic property of the calculated invariant, which, I 
believe, was not suspected. 
0 It will at once be recognized that “ No. 19 ” given in Mr. Cayley’s Second Memoir upon Quantics is iden- 
tical with the J of this memoir, whence it follows from Mr. Cayley’s equation (No. 26)= (No. 19) 2 — 1152 
No. 26, that K=9 (No. 25). Thus abstraction made of a mere numerical factor, Mr. Cayley and myself agree 
upon perfectly distinct grounds in recognizing K and L as the true simplest invariants of their respective 
degrees, an accordance as satisfactory as it was unexpected, and which must be considered as setting at rest the 
question of what should be deemed the, so to say, staple invariants of the Binary Quintic. 
