658 
PROFESSOR SYLVESTER ON THE REAL 
upwards. Thus M. Hermite, by a method not introducing the notion of continuity, has 
found one of the twenty-fourth degree, which he has been so obliging as to communi- 
cate to me, viz. (Dj— 5AJ 2 ) 2 +(9D— 25A 2 )J 2 , where D 1 =16J 3 +25AJ 2 ; and D is his 
discriminant, which I cannot safely attempt to express in terms of x, y for want of 
a certain knowledge of the arithmetical relations between his A, J 2 , J 3 D, and my own 
J, K, L ; but were this transformation effected, the curve so represented must, ex neces- 
sitate, pass clear between the triangle and sector above referred to, or else the invariant in 
question could not be a criterion. I have ascertained without difficulty that it passes 
through the origin and represents one of the principal species of Newton’s diverging 
parabolas. 
(91) The curve which we have been discussing will, on reference to Plucker’s ‘ Alge- 
braischen Curven,’ p. 193, be seen to belong to his sixteenth species of curves of the 
fourth order having two double points ; but as in reality one of these is tantamount to 
the union of two, it may be considered as having three, the maximum possible number 
of such points, and consequently comes under the operation of Clebsch’s rule, given in 
the last Number of Crelle’s Journal, and accordingly its coordinates have been seen to 
be rational functions of a single variable. The equation connecting x, y may of course 
be obtained by means of a simple and obvious substitution operated upon the G of 
art. 41, or it may be found directly by writing 
x + \ 1 y— 1 2<p + 3' 
8 ^ <p 4 + <p 3 4 — 7i ~ ~f 3 + <p 2 ’ 
whence we obtain 
p+p - |=°, a ) 
2f + 3<f,+j=0 (2) 
Calling <p t (p 2 the two roots of equation (2), making 
and substituting the values of the symmetric functions of <p x , (p 2 found from the same 
equation, we obtain without difficulty 
^__|^_8^+36r^+16r-27i 3 =0 
for the equation in question. The curve thus denoted I propose to call the Bicorn. 
Its figure is given in Plate XXV., in which j? are taken at right angles, but they may 
of course be supposed to be inclined at any angle whatever. If we now assume at 
pleasure any two new axes U, V in the place of the Bicorn, the coordinates u, v will be 
always respectively proportional to two invariants of the twelfth order of the given 
quintic, whose particular forms will depend upon the positions of the two new axes so 
taken. If one of these axes, say that of u, be made coincident with the axis of v will 
be proportional to JD, and u to some other invariant of the twelfth degree. When 
