AND IMAGINARY ROOTS OR EQUATIONS. 
659 
this is the case, then in general v, as u travels from one end of infinity to the other, 
will sometimes have four, and sometimes two, or else sometimes two and sometimes no 
real values, as will be obvious by inspection of the figure. There is, however, one 
direction of the axis of v which will cause v in all cases to have two, and only two real 
values. This direction is that of the line joining the two cusps. At the node-cusp, for 
which <p= co, £=0, ?7=0; at the other cusp, for which <p= — f, §= — ^ 7 -, jj= — - 3 -. 
Hence the equation of the joining line is 9£— 8;j =0. Now ^|= — ^ J^ = 256' •^■ ence 
along this line 9L+JK=0 ; and consequently, if the axis of v be taken parallel to this 
line and passing through the origin, whilst u is proportional to 9L+JK, v will be pro- 
portional to JD ; and thus we see that for every value of 9L+JK, which is Hermite’s 
J 3 (see foot-note ( 34 )( e )), D at the amphigenous surface (i. e. when G=0, and therefore 
when Hermite’s 1=0) will always have two, and only two real values. This perfectly 
agrees with M. Hermite’s conclusion ( 71 ), and in an unexpected manner confirms the 
correctness of the concordance established, in the foot-note cited, between his J 3 and 
my J, K, L. Had M. Hermite employed any duodecimal invariant whatever other 
than J 3 , a mere inspection of the Bicorn shows that a similar conclusion could not have 
obtained. 
(92) The intersections of the curve whose equation is written in the preceding article 
with infinity evidently lie in the lines fj 3 = 0, v \ — £ = 0. This latter is the equation 
to a line parallel to the asymptote which touches the highest and lowest of the four 
branches of the curve, and corresponds to the value — 1 of <p. Thus we see that there 
is a point of inflexion corresponding to the point at infinity at which the second and 
third branches of the Bicorn may be conceived to unite. It is easy to show that the 
Bicorn has no double tangent ; for we have seen that 
and consequently the values of <p corresponding to the two supposed points of contact 
may be regarded as the two roots <p 2 of the equation <p 2 +2<p + 2A=0, and we shall have 
_- 2fa+3 2fa + 3 / 2 _ 2 \ 
fi + flVl + Vt \tf+P 3 i <p2 + ?2/’ 
i. ^-(2^+3)(^+^)+(2^+3)(^+^)=(^+^)-(^+^) 3 , 
or 
4\.(— 2) + 4*+3(4— 2x)+6(— 2(4— 4x)-f(4-2x)) = 0, 
or 
(_ 8-{-4—6-f-8— 2>+12 — 6 — 8+4=0, 
i. c. — 4x+2 = 0, <p 2 +2<p+l=0, 
and the two values of coincide, contrary to hypothesis. 
It is also easy to find its class ; for when ^ corresponds to any point in which the 
( n ) Lemma 3, p. 202, Cambridge and Dublin Journal, vol. ix. 
MDCCCLXIV. 4 T 
