642 
PROFESSOR SYLVESTER ON THE REAL 
(66) The assertion that the whole of facultative space is divisible into three regions, in 
strictness requires a slight modification. It is obvious that the plane of D itself cannot 
be said to belong to any of the regions ; and in order to make our theory quite complete, 
so as to furnish criteria applicable to equations having equal roots, and to enable us to 
distinguish between the case of the unequal roots being all three real, or two imaginary 
and one real, we must examine what takes place in this plane, and under what circum- 
stances a passage from one point of it to another will or may be accompanied with a 
change of character in the roots. 
If the roots of/(#)=0 are supposed to be a , «, c, d, e, where c, d, e are unequal, 
on varying the constants of fx in such a manner that the variation of the discrimi- 
nant D is zero, the two equal roots a, a will remain equal. Now in general we have 
» if this, under the particular supposition made, continued to obtain, 
da would have two distinct values, and the two equal roots would cease to continue to 
be equal, contrary to hypothesis. Hence we see that D=0, ^D=0 necessarily implies 
£/■(«) =0( 55 ), and consequently lf(a-\-da) is no longer Ifa, but If'ada; so that we obtain 
da— 0, or da— — 
2 Ifa 
and no change of character in the five roots results. If, however, 
the original roots are a, a, c, c, e, then, as shown in the general case, oc will have two 
distinct values, which will be both real or both imaginary. Accordingly we see that in 
( t5 )( a ) This is a somewhat curious theorem (whether new or otherwise I know not) thus incidentally established 
in the text, viz. that if D(/) represent the discriminant of/, and if D(/)=0 and SD(f)=0, then when/=0 we 
must have $(/)=0. The very simplest example that can be chosen will serve to illustrate this proposition. Let 
f=aoc?+2bxy+cif. 
Suppose 
D(/)=ac — b 2 = 0, 
and also 
SD(f)=aSc+c$a — 2bSb=0, 
we have 
8(f)=x 2 Sa,+2xy8b+y 2 $c. 
Now if/=0 we may write x=b, y=—a, and Sf becomes 
b 2 Sa — 2abSb + a?Sc 
= b 2 Sa — 2abSb + 2abSb— acSa 
=(b 2 — ac)Sa—0, 
according to the theorem. 
If we make/=(a7, 1)", D we know becomes a syzygetic function off and /'^meaning by the latter Hence 
since £D vanishes when/r=0, D=0, and 8f(x)=0, we learn that S(D) is a syzygetic function of (/,/', Sf). 
The theorem thus stated easily admits of extension to the higher variations of D, and so extended takes I 
believe the following form : 
£‘(D)= a syzygetic function of (f, f, f", . . . ./*’, of). 
( b ) Professor Cayley has since informed me that the theorem in ( 5S ) ( a ), about whose originality I was in doubt, 
will be found in Schlafli’s £ De Eliminatione.’ This is not the first unconscious plagiarism I have been guilty of 
towards this eminent man, whose friendship I am proud to claim. A much more glaring case occurs in a note 
by me in the £ Comptes Rendus,’ on the twenty- seven straight lines of cubic surfaces, where I believe I have 
followed (like one walking in his sleep), down to the very nomenclature and notation, the substance of a por- 
tion of a paper inserted by Schxafli in the £ Mathematical Journal,’ which bears my name as one of the editors 
upon its face ! 
