654 
PROFESSOR SYLVESTER ON THE REAL 
Received December 8, 1864. 
Note on the arbitrary constant which appears in one of the criteria for distinguishing the 
case of four from that of no imaginary roots , and on the curve whose coordinates ex- 
press the limiting relations of all the octodecimal invariants of a binary quintic, &c. 
(85) The appearance of an arbitrary constant in a criterion is a circumstance so unex- 
ampled and remarkable that I have thought it desirable to give a more complete, or at 
least a more palpable proof of the validity of the substitution of A+^JD for A than 
that furnished in the foregoing text, where some indistinctness arises from the diffi- 
culty of raising up in the mind a clear conception of the form of the amphigenous 
surface, and the two portions of space which it separates. That difficulty is entirely 
obviated, and the theory rendered palpable to the senses by the following investigation, 
where the problem is so handled as to involve the contemplation of two dimensions only 
of space. We have in general 
D=P— 128K, A=2048L— J 3 , 
and at the amphigenous surface (see art. 57) 
Let 
Then 
K 0 + 6 L 1 
J 2 “ (6 + 4)6*’ J 3_ (0 + 4)0 3 * 
. , D A 
0=4:<p, y= j 2 , x=j- 3 - 
y =1-128 
6 + 6 
(6+4)6*~ 
8<p + l2 _ (p + 2) 2 (p-3 ) 
?% + !)“ ?(<P + 1 ) ’ 
and consequently 
_ 2048 _ 8 
x ~ 1 + (0 + 4)0 3- f+<g 
4(<p + 2)(4<p + 3) v 
-(<p + 2)(f-f + 2v-4) . 
8(4<p + 3K f + 2 <p 
>% + !)* 2 ■ 
x, y may be considered as the coordinates (inclined to each other at any angle) of a curve 
of the fourth order, whose form, so far as is essential to the object in view, I proceed to 
determine. It is obvious, furthermore, that this curve will be a section of the amphi- 
genous surface made by the plane J=l. 
(86) This curve will be seen to consist of four branches, coming together in pairs or 
two cusps, so as to form two distinct horns( 65 ). For when <p= oo , or <p=— f, ty, will 
( 65 )( a ) Since f + 0, 
we see at once, from Descartes’s rule, that <p can never have more than two real values to one of or con- 
sequently of x, and consequently there can only he two values of y to each of x. 
(”) When J=0, the cusp of the left-hand horn and the two points of intersection of the dexter horn with 
the axis of L coincide at the origin ; the upper branch of the latter and the linear of the former become the 
lower and upper parts of the axis of D, whilst the lower and upper branches of the same respectively become 
the left, and right-hand branches of the semicubical parabola 27.2 s2 L 2 = — D 3 . 
