660 
PEOFESSOE SYLYESTEE ON THE EEAL 
curve is met by a tangent drawn from the point whose £, ^ coordinates are a, b, we have 
but 
hence 
hence 
\ */_j. 
a )=0 
+r 
l= 2 2=-(^+ 2 ?); 
+ lx if + 2) +(^+2y)«+i=0 ; 
l +2ap 3 +&p 2 -|-l = 0 ; 
and <p having four values, four tangents (real or imaginary) can be drawn to the Bicorn 
from every point in its plane. It is thus of the fourth order, fourth class, possesses a 
common cusp and a cusp-node, no double tangent, and one point of inflexion at infinity. 
These results accord with those given by Plucker (Algebraischen Curven, p. 222). 
(93) The canonical form of the equation to the Bicorn is (i^+<f) 2 +P2' 3 = 0, as seen 
in Plucker, p. 193, where ]?=0, r— 0, ^=0 will obviously be the equations to the 
tangent at the node-cusp, to the tangent at the common cusp, and to the line of junction 
of the two cusps. This leads to a remarkable transformation of the invariant G of 
art. (41). Thus we may write _p=|, ^=^(9^—8^); and to find r, we must draw the 
tangent to the lower cusp, for which cp= — f, which gives 
256 _ 32 <k_ 15 /72 , 
27 » *?— 3 ’ d £— 16 ( J; 
consequently we may write r=X(144;?— 1351+256), and then proceed to satisfy, by 
assigning suitable values to X, v, the identity 
(x(144,|-135| a +256|)+^(8»-94)t+f‘ s l(8‘I- 9 l) 3 
■|-8 ! j , |+36 s | ! +16| J -27f)=» . 2”G. 
On performing the necessary calculations it will be found that 
, 1 __l 1 
^ 2 12 ’ ^ 2 6 ’ V — 2 12 " 
Hence we see that J 3 G may be expressed under the form (LLj+cJ^+eLJg, where 
is a new duodecimal invariant, and c, e are two known numbers ; in fact 
J 3 G=(L(18JK+135L 2 -J 3 L)+(JK+9L) 2 ) 2 +64L(JK+9L) 3 . 
I am indebted to my friend Dr. Hirst for these references to the immortal work of 
Plucker. 
(94) The existence has been demonstrated of a linear asymptote which is a tangent 
( 72 ) I find, by a calculation which offers no difficulty, that the value of <p at the point where this tangent cuts 
the curve will be given by the equation 
-256<p 4 -256<p 3 +288f+432cp+135=0; 
and taking away the factor (4p+3) 3 which belongs to the cusp, there [remains p=f, which corresponds to a 
point in the lower branch of the superior horn. 
